Calculus

Problem 33401

Evaluate the integral: $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ .

See Solution

Problem 33402

Find the derivative of the function $f(x)=\frac{2 x^{5}-3 x^{4}-7 x^{3}}{x^{4}}$ . What is $f^{\prime}(t)$ ?

See Solution

Problem 33403

True or False: Does the integral $\int_{2}^{\infty} \frac{1}{x^{3} \ln (x)} dx$ converge?

See Solution

Problem 33404

Find the intervals when the acceleration of a particle, defined by $x(t)=2 t^{4}-16 t^{3}$ for $t \geq 0$ , is negative.

See Solution

Problem 33405

Find the derivative $y^{\prime}$ if $y=\sqrt{\ln \sqrt{t}}$ . Options: a) $\frac{1}{4 t \sqrt{\ln (t)}}$ b) $\frac{1}{2 \sqrt{\ln \sqrt{t}}}$ c) $\frac{1}{4 t \sqrt{\ln \sqrt{t}}}$ d) $\frac{1}{2 t \sqrt{\ln (t)}}$ e) $\frac{1}{2 t \sqrt{\ln \sqrt{t}}}$

See Solution

Problem 33406

Find the intervals where the particle moves right for $x(t)=3 t^{3}-9 t^{2}-72 t$ , $t \geq 0$ .

See Solution

Problem 33407

Find the derivative of $f(x) = \left(\sin(4x + 3)\right)^{x^3 - 2}$ using logarithms.

See Solution

Problem 33408

Find the derivative $y^{\prime}$ if $y=\sqrt{\ln \sqrt{t}}$ . Options: a) $\frac{1}{4 t \sqrt{\ln (t)}}$ , b) $\frac{1}{2 \sqrt{\ln \sqrt{t}}}$ , c) $\frac{1}{4 t \sqrt{\ln \sqrt{t}}}$ , d) $\frac{1}{2 t \sqrt{\ln (t)}}$ , e) $\frac{1}{2 t \sqrt{\ln \sqrt{t}}}$ .

See Solution

Problem 33409

Find the critical points of the function $f(x)=x e^{x^{2}}$ . Options: a. none, b. $\frac{1}{3 \sqrt{e}}$ , c. $e^{-\frac{1}{2}}$ , d. $-\frac{1}{2}$ .

See Solution

Problem 33410

Find the values of $x$ where the rate of change of $y=\frac{1}{3} x^{3}-3 x$ equals that of $x$ . Options: $\pm 1$ , $\pm 2$ , $\pm 1.5$ , $\pm 2.5$ .

See Solution

Problem 33411

Find the tangent line approximation for $f(3.9)$ given $f(4)=-9$ and $f'(4)=3$ .

See Solution

Problem 33412

Evaluate the integral from 1 to e of $\frac{2 \ln x + 3}{x} \, dx$ : Select one 3, None, 2, or 5.

See Solution

Problem 33413

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . Options: 0, 1, $e^{-1}$ , none, e.

See Solution

Problem 33414

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e$ , $e^{-1}$ , none, 1, 0.

See Solution

Problem 33415

Evaluate the integral from 1 to 2: $\int_{1}^{2} x^{x}(1+\ln (x)) d x=$ Select one: $0$ , 2, none, $3$ , 10.

See Solution

Problem 33416

Find the derivative $y^{\prime}$ of the function $y=(\sqrt{t})^{t}$ .

See Solution

Problem 33417

Find the derivative of $y=(\sqrt{t})^{t}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33418

Evaluate the integral from 3 to 4: $\int_{3}^{4} x^{x}(1+\ln (x)) d x=$ . Choose from: 5, 229, 221, none, 7.

See Solution

Problem 33419

Calculate the integral from 3 to 4 of $x^{x}(1+\ln (x))$ dx. What is the result? Options: none, 221, 7, 5, 229.

See Solution

Problem 33420

Find the derivative of the function $y=x^{\ln (x)}$ for $x>0$ .

See Solution

Problem 33421

Evaluate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ .

See Solution

Problem 33422

Find the derivative $y^{\prime}$ if $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

See Solution

Problem 33423

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{n}}{5^{n^{2}}}$ converges or diverges.

See Solution

Problem 33424

Calculate the integral from 1 to 2 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choices: 1, 0, none, 3, 2.

See Solution

Problem 33425

Evaluate the integral from 2 to e: $\int_{2}^{e} \frac{1+\ln (x)}{x \ln (x)} d x=$ . Select the correct answer.

See Solution

Problem 33426

Find the derivative $y'$ , if $y=\log_{2}(8 t^{\ln 2})$ .

See Solution

Problem 33427

Find the derivative of $y=\sin \left(x^{x}\right)$ , i.e., $y^{\prime}=$ ? Select the correct option.

See Solution

Problem 33428

Evaluate the integral from $e^2$ to $e^3$ of $\frac{1+\ln(x)}{x \ln(x)} \, dx$ . What is the result?

See Solution

Problem 33429

Find the derivative $y'$ , where $y=(x+1)^{x}$ . Choose the correct option.

See Solution

Problem 33430

Find the derivative $y'$ , where $y=x \sin(\log_7 x)$ . Select the correct expression for $y'$ .

See Solution

Problem 33431

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(10)^{n}(n+5) !}{(2 n+1) !}$ . Choose from: a. $\frac{1}{2}$ b. 0 c. 10 d. $\infty$ e. 5

See Solution

Problem 33432

Calculate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ and choose the correct answer.

See Solution

Problem 33433

Find the derivative $y'$ , if $y=3 \log _{8}(\log _{2} t)$ .

See Solution

Problem 33434

Find the tangent line to $f$ at $x=1$ and use it to approximate $f(0.95)$ . Given $f(1)=7$ and $f'(x)=-3x+5$ .

See Solution

Problem 33435

Evaluate the integral: $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$

See Solution

Problem 33436

Find the derivative of $y=\sin \left(x^{x}\right)$ , i.e., calculate $y^{\prime}$ .

See Solution

Problem 33437

Find the derivative of $y=(\sqrt{t})^{t}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33438

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Choose from: $e^{-1}$ , 1, 0, e, none.

See Solution

Problem 33439

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options.

See Solution

Problem 33440

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Choices: $\frac{t}{\ln 2}$ , $\frac{\ln 8}{t}$ , none, $3+\ln t$ , $\frac{1}{t}$ .

See Solution

Problem 33441

Calculate the integral from 1 to 2 of $x^{x}(1+\ln (x))$ dx. What is the value? Options: 0, 3, 2, 1, none.

See Solution

Problem 33442

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list.

See Solution

Problem 33443

Evaluate the integral: $\int \frac{d x}{x \log _{10} x}$ . Select the correct answer from the options given.

See Solution

Problem 33444

Find the derivative $y'$ , where $y=x \sin(\log_{7} x)$ . Options include various expressions involving $\sin$ and $\cos$ .

See Solution

Problem 33445

Evaluate the integral from 2 to e of (1 + ln(x)) / (x ln(x)) dx. What is the result?

See Solution

Problem 33446

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Options: e, 2, 0, none, 1.

See Solution

Problem 33447

Calculate the integral $\int_{1}^{2} x^{x}(1+\ln (x)) d x$ .

See Solution

Problem 33448

Evaluate the integral $\int \frac{d x}{x(\log_{8} x)^{2}}$ and choose the correct answer.

See Solution

Problem 33449

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e$ , $e^{-1}$ , none, 0, 1.

See Solution

Problem 33450

Calculate the integral from 3 to 4 of $x^{x}(1+\ln (x))$ with respect to $x$ .

See Solution

Problem 33451

Find the derivative of $y=(x+1)^{x}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33452

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}+1}$ .

See Solution

Problem 33453

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+1}$ using comparison tests.

See Solution

Problem 33454

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list provided.

See Solution

Problem 33455

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}$ . Choices: a. 4 b. $\infty$ c. 0 d. 1 e. 2

See Solution

Problem 33456

Evaluate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choose from: 23, none, 1, 21, 5.

See Solution

Problem 33457

Find the derivative $y^{\prime}$ of $y=t \log _{3}(e^{(\sin t)(\ln 3)})$ .

See Solution

Problem 33458

Find the derivative of $y=(\ln x)^{\ln x}$ . What is $y'$ ?

See Solution

Problem 33459

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choose: 21, none, 1, 23, 5.

See Solution

Problem 33460

Evaluate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Options: 1, 5, 21, none, 23.

See Solution

Problem 33461

Determine the convergence of the series $\sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2}-1}}$ . Options: a. none b. diverges with $\sum_{n=2}^{\infty} \frac{1}{n}$ c. diverges with $\sum_{n=2}^{\infty} \frac{1}{\sqrt{n^{2}+1}}$ d. converges with $\sum_{n=2}^{\infty} \frac{1}{n^{2}}$ e. converges with $\sum_{n=2}^{\infty} \frac{1}{n}$ .

See Solution

Problem 33462

Evaluate the integral: $I = \int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ . Choose the correct answer from the options.

See Solution

Problem 33463

Find the integral of $\frac{1}{x \log_{10} x}$ and choose the correct answer from the options given.

See Solution

Problem 33464

Find the derivative $y^{\prime}$ if $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

See Solution

Problem 33465

Find the derivative of $y=(\sqrt{t})^{t}$ . What is $y^{\prime}$ ?

See Solution

Problem 33466

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the result? Options: 23, none, 21, 5, 1.

See Solution

Problem 33467

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose from: $2 x^{\ln (x)-1} \ln (x)$ , $2 x^{\ln (x)} \ln (x)$ , $x^{\ln (x)-1} \ln (x)$ , none, $x^{\ln (x)} \ln (x)$ .

See Solution

Problem 33468

Find the integral of $x^{2 x}(1+\ln x) \, dx$ . What is the result?

See Solution

Problem 33469

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Options: $3+\ln t$ , $\frac{1}{t}$ , $\frac{t}{\ln 2}$ , $\frac{\ln 8}{t}$ , none.

See Solution

Problem 33470

Find the derivative $y'$ , where $y = (x+1)^x$ . Choose from: $(x+1)^{x}(\frac{1}{x+1}+\ln(x+1))$ or others.

See Solution

Problem 33471

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ .

See Solution

Problem 33472

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ dx. What is the result? Options: 21, 5, 23, 1.

See Solution

Problem 33473

Calculate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ .

See Solution

Problem 33474

Find the derivative $y^{\prime}$ of $y=3 \log _{8}(\log _{2} t)$ . Choices include $\frac{3}{t(\ln t)(\ln 2)}$ and others.

See Solution

Problem 33475

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct result.

See Solution

Problem 33476

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ .

See Solution

Problem 33477

Find the derivative $y^{\prime}$ if $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

See Solution

Problem 33478

Determine if the series $\sum_{n=1}^{\infty} \frac{7+10 \cos n}{n^{4}}$ diverges or converges.

See Solution

Problem 33479

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ .

See Solution

Problem 33480

Calculate the integral from 3 to 4 of $x^{x}(1+\ln(x)) \, dx$ . What is the result? Options: none, 7, 221, 229, 5.

See Solution

Problem 33481

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ .

See Solution

Problem 33482

Find the integral of the function $x^{x}(1+\ln (x))$ .

See Solution

Problem 33483

Evaluate the integral from 3 to 4 of $x^{x}(1+\ln (x))$ . What is the result? Select from: 229, 5, 221, none, 7.

See Solution

Problem 33484

Find the derivative of $y=(x+1)^{x}$ . What is $y^{\prime}$ ? Choose the correct option.

See Solution

Problem 33485

Find the derivative $y^{\prime}$ of $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

See Solution

Problem 33486

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e^{-1}$ , none, 0, 1.

See Solution

Problem 33487

Evaluate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the result? Options: 23, 21, 5, none, 1.

See Solution

Problem 33488

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: none, 1, 0, $e^{-1}$ , $e$ .

See Solution

Problem 33489

Find the derivative of $y=(\ln x)^{\ln x}$ . What is $y^{\prime}=$ ?

See Solution

Problem 33490

Find the derivative $y^{\prime}$ if $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

See Solution

Problem 33491

Evaluate the integral: $\int \frac{d x}{x \log_{10} x}$ .

See Solution

Problem 33492

Find the derivative $y'$ if $y = (x+1)^x$ .

See Solution

Problem 33493

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choose: none, 1, 5, 23, 21.

See Solution

Problem 33494

Find the integral: $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ . Choose the correct result from the options given.

See Solution

Problem 33495

Find the derivative $y^{\prime}$ of the function $y=(x+1)^{x}$ .

See Solution

Problem 33496

Find the derivative $y^{\prime}$ of $y=x \sin(\log_{7} x)$ . Choose from the options provided.

See Solution

Problem 33497

Find the derivative $y'$ of $y = (x+1)^x$ . Choose the correct expression for $y'$ .

See Solution

Problem 33498

Find the derivative of $y=\sin \left(x^{x}\right)$ , i.e., $y^{\prime}=$ ?

See Solution

Problem 33499

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options provided.

See Solution

Problem 33500

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ dx. What is the result? Options: 1, 23, none, 5, 21.

See Solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337

Calculus

Problem 33401

Evaluate the integral: ∫sec⁡(3x)tan⁡(3x)2+sec⁡(3x)dx\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x∫2+sec(3x)sec(3x)tan(3x)​dx.

Problem 33402

Find the derivative of the function f(x)=2x5−3x4−7x3x4f(x)=\frac{2 x^{5}-3 x^{4}-7 x^{3}}{x^{4}}f(x)=x42x5−3x4−7x3​. What is f′(t)f^{\prime}(t)f′(t)?

Problem 33403

True or False: Does the integral ∫2∞1x3ln⁡(x)dx\int_{2}^{\infty} \frac{1}{x^{3} \ln (x)} dx∫2∞​x3ln(x)1​dx converge?

Problem 33404

Find the intervals when the acceleration of a particle, defined by x(t)=2t4−16t3x(t)=2 t^{4}-16 t^{3}x(t)=2t4−16t3 for t≥0t \geq 0t≥0, is negative.

Problem 33405

Problem 33406

Find the intervals where the particle moves right for x(t)=3t3−9t2−72tx(t)=3 t^{3}-9 t^{2}-72 tx(t)=3t3−9t2−72t, t≥0t \geq 0t≥0.

Problem 33407

Find the derivative of f(x)=(sin⁡(4x+3))x3−2f(x) = \left(\sin(4x + 3)\right)^{x^3 - 2}f(x)=(sin(4x+3))x3−2 using logarithms.

Problem 33408

Problem 33409

Find the critical points of the function f(x)=xex2f(x)=x e^{x^{2}}f(x)=xex2. Options: a. none, b. 13e\frac{1}{3 \sqrt{e}}3e​1​, c. e−12e^{-\frac{1}{2}}e−21​, d. −12-\frac{1}{2}−21​.

Problem 33410

Find the values of xxx where the rate of change of y=13x3−3xy=\frac{1}{3} x^{3}-3 xy=31​x3−3x equals that of xxx. Options: ±1\pm 1±1, ±2\pm 2±2, ±1.5\pm 1.5±1.5, ±2.5\pm 2.5±2.5.

Problem 33411

Find the tangent line approximation for f(3.9)f(3.9)f(3.9) given f(4)=−9f(4)=-9f(4)=−9 and f′(4)=3f'(4)=3f′(4)=3.

Problem 33412

Evaluate the integral from 1 to e of 2ln⁡x+3x dx\frac{2 \ln x + 3}{x} \, dxx2lnx+3​dx: Select one 3, None, 2, or 5.

Problem 33413

Find the derivative of y=xln⁡(ln⁡(x))y=x^{\ln (\ln (x))}y=xln(ln(x)) at x=ex=ex=e. Options: 0, 1, e−1e^{-1}e−1, none, e.

Problem 33414

Find the derivative of y=xln⁡(ln⁡(x))y=x^{\ln (\ln (x))}y=xln(ln(x)) at x=ex=ex=e. What is the value? Options: eee, e−1e^{-1}e−1, none, 1, 0.

Problem 33415

Evaluate the integral from 1 to 2: ∫12xx(1+ln⁡(x))dx=\int_{1}^{2} x^{x}(1+\ln (x)) d x=∫12​xx(1+ln(x))dx= Select one: 000, 2, none, 333, 10.

Problem 33416

Find the derivative y′y^{\prime}y′ of the function y=(t)ty=(\sqrt{t})^{t}y=(t​)t.

Problem 33417

Find the derivative of y=(t)ty=(\sqrt{t})^{t}y=(t​)t. What is y′=y^{\prime}=y′=?

Problem 33418

Evaluate the integral from 3 to 4: ∫34xx(1+ln⁡(x))dx=\int_{3}^{4} x^{x}(1+\ln (x)) d x=∫34​xx(1+ln(x))dx=. Choose from: 5, 229, 221, none, 7.

Problem 33419

Calculate the integral from 3 to 4 of xx(1+ln⁡(x))x^{x}(1+\ln (x))xx(1+ln(x)) dx. What is the result? Options: none, 221, 7, 5, 229.

Problem 33420

Find the derivative of the function y=xln⁡(x)y=x^{\ln (x)}y=xln(x) for x>0x>0x>0.

Problem 33421

Evaluate the integral from e2e^{2}e2 to e3e^{3}e3 of 1+ln⁡(x)xln⁡(x) dx\frac{1+\ln (x)}{x \ln (x)} \, dxxln(x)1+ln(x)​dx.

Problem 33422

Find the derivative y′y^{\prime}y′ if y=tlog⁡3(e(sin⁡t)(ln⁡3))y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)y=tlog3​(e(sint)(ln3)).

Problem 33423

Determine if the series ∑n=1∞nn5n2\sum_{n=1}^{\infty} \frac{n^{n}}{5^{n^{2}}}∑n=1∞​5n2nn​ converges or diverges.

Problem 33424

Calculate the integral from 1 to 2 of xx(1+ln⁡(x)) dxx^{x}(1+\ln (x)) \, dxxx(1+ln(x))dx. What is the result? Choices: 1, 0, none, 3, 2.

Problem 33425

Evaluate the integral from 2 to e: ∫2e1+ln⁡(x)xln⁡(x)dx=\int_{2}^{e} \frac{1+\ln (x)}{x \ln (x)} d x=∫2e​xln(x)1+ln(x)​dx=. Select the correct answer.

Problem 33426

Find the derivative y′y'y′, if y=log⁡2(8tln⁡2)y=\log_{2}(8 t^{\ln 2})y=log2​(8tln2).

Problem 33427

Find the derivative of y=sin⁡(xx)y=\sin \left(x^{x}\right)y=sin(xx), i.e., y′=y^{\prime}=y′=? Select the correct option.

Problem 33428

Evaluate the integral from e2e^2e2 to e3e^3e3 of 1+ln⁡(x)xln⁡(x) dx\frac{1+\ln(x)}{x \ln(x)} \, dxxln(x)1+ln(x)​dx. What is the result?

Problem 33429

Find the derivative y′y'y′, where y=(x+1)xy=(x+1)^{x}y=(x+1)x. Choose the correct option.

Problem 33430

Find the derivative y′y'y′, where y=xsin⁡(log⁡7x)y=x \sin(\log_7 x)y=xsin(log7​x). Select the correct expression for y′y'y′.

Problem 33431

Problem 33432

Calculate the integral from e2e^{2}e2 to e3e^{3}e3 of 1+ln⁡(x)xln⁡(x) dx\frac{1+\ln (x)}{x \ln (x)} \, dxxln(x)1+ln(x)​dx and choose the correct answer.

Problem 33433

Find the derivative y′y'y′, if y=3log⁡8(log⁡2t)y=3 \log _{8}(\log _{2} t)y=3log8​(log2​t).

Problem 33434

Find the tangent line to fff at x=1x=1x=1 and use it to approximate f(0.95)f(0.95)f(0.95). Given f(1)=7f(1)=7f(1)=7 and f′(x)=−3x+5f'(x)=-3x+5f′(x)=−3x+5.

Problem 33435

Evaluate the integral: ∫2log⁡2(x−1)(x−1)dx\int \frac{2 \log _{2}(x-1)}{(x-1)} d x∫(x−1)2log2​(x−1)​dx

Problem 33436

Find the derivative of y=sin⁡(xx)y=\sin \left(x^{x}\right)y=sin(xx), i.e., calculate y′y^{\prime}y′.

Problem 33437

Find the derivative of y=(t)ty=(\sqrt{t})^{t}y=(t​)t. What is y′=y^{\prime}=y′=?

Problem 33438

Find the derivative of y=xln⁡(ln⁡(x))y=x^{\ln (\ln (x))}y=xln(ln(x)) at x=ex=ex=e. What is the value? Choose from: e−1e^{-1}e−1, 1, 0, e, none.

Problem 33439

Evaluate the integral: ∫dxx(log⁡8x)2\int \frac{d x}{x\left(\log _{8} x\right)^{2}}∫x(log8​x)2dx​. Choose the correct answer from the options.

Problem 33440

Find the derivative y′y^{\prime}y′ if y=log⁡2(8tln⁡2)y=\log _{2}(8 t^{\ln 2})y=log2​(8tln2). Choices: tln⁡2\frac{t}{\ln 2}ln2t​, ln⁡8t\frac{\ln 8}{t}tln8​, none, 3+ln⁡t3+\ln t3+lnt, 1t\frac{1}{t}t1​.

Problem 33441

Calculate the integral from 1 to 2 of xx(1+ln⁡(x))x^{x}(1+\ln (x))xx(1+ln(x)) dx. What is the value? Options: 0, 3, 2, 1, none.

Evaluate the integral: $\int \frac{\sec (3 x) \tan (3 x)}{2+\sec (3 x)} d x$ .

Find the derivative of the function $f(x)=\frac{2 x^{5}-3 x^{4}-7 x^{3}}{x^{4}}$ . What is $f^{\prime}(t)$ ?

True or False: Does the integral $\int_{2}^{\infty} \frac{1}{x^{3} \ln (x)} dx$ converge?

Find the intervals when the acceleration of a particle, defined by $x(t)=2 t^{4}-16 t^{3}$ for $t \geq 0$ , is negative.

Find the intervals where the particle moves right for $x(t)=3 t^{3}-9 t^{2}-72 t$ , $t \geq 0$ .

Find the derivative of $f(x) = \left(\sin(4x + 3)\right)^{x^3 - 2}$ using logarithms.

Find the critical points of the function $f(x)=x e^{x^{2}}$ . Options: a. none, b. $\frac{1}{3 \sqrt{e}}$ , c. $e^{-\frac{1}{2}}$ , d. $-\frac{1}{2}$ .

Find the values of $x$ where the rate of change of $y=\frac{1}{3} x^{3}-3 x$ equals that of $x$ . Options: $\pm 1$ , $\pm 2$ , $\pm 1.5$ , $\pm 2.5$ .

Find the tangent line approximation for $f(3.9)$ given $f(4)=-9$ and $f'(4)=3$ .

Evaluate the integral from 1 to e of $\frac{2 \ln x + 3}{x} \, dx$ : Select one 3, None, 2, or 5.

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . Options: 0, 1, $e^{-1}$ , none, e.

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e$ , $e^{-1}$ , none, 1, 0.

Evaluate the integral from 1 to 2: $\int_{1}^{2} x^{x}(1+\ln (x)) d x=$ Select one: $0$ , 2, none, $3$ , 10.

Find the derivative $y^{\prime}$ of the function $y=(\sqrt{t})^{t}$ .

Find the derivative of $y=(\sqrt{t})^{t}$ . What is $y^{\prime}=$ ?

Evaluate the integral from 3 to 4: $\int_{3}^{4} x^{x}(1+\ln (x)) d x=$ . Choose from: 5, 229, 221, none, 7.

Calculate the integral from 3 to 4 of $x^{x}(1+\ln (x))$ dx. What is the result? Options: none, 221, 7, 5, 229.

Find the derivative of the function $y=x^{\ln (x)}$ for $x>0$ .

Evaluate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ .

Find the derivative $y^{\prime}$ if $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

Determine if the series $\sum_{n=1}^{\infty} \frac{n^{n}}{5^{n^{2}}}$ converges or diverges.

Calculate the integral from 1 to 2 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choices: 1, 0, none, 3, 2.

Evaluate the integral from 2 to e: $\int_{2}^{e} \frac{1+\ln (x)}{x \ln (x)} d x=$ . Select the correct answer.

Find the derivative $y'$ , if $y=\log_{2}(8 t^{\ln 2})$ .

Find the derivative of $y=\sin \left(x^{x}\right)$ , i.e., $y^{\prime}=$ ? Select the correct option.

Evaluate the integral from $e^2$ to $e^3$ of $\frac{1+\ln(x)}{x \ln(x)} \, dx$ . What is the result?

Find the derivative $y'$ , where $y=(x+1)^{x}$ . Choose the correct option.

Find the derivative $y'$ , where $y=x \sin(\log_7 x)$ . Select the correct expression for $y'$ .

Calculate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ and choose the correct answer.

Find the derivative $y'$ , if $y=3 \log _{8}(\log _{2} t)$ .

Find the tangent line to $f$ at $x=1$ and use it to approximate $f(0.95)$ . Given $f(1)=7$ and $f'(x)=-3x+5$ .

Evaluate the integral: $\int \frac{2 \log _{2}(x-1)}{(x-1)} d x$

Find the derivative of $y=\sin \left(x^{x}\right)$ , i.e., calculate $y^{\prime}$ .

Find the derivative of $y=(\sqrt{t})^{t}$ . What is $y^{\prime}=$ ?

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Choose from: $e^{-1}$ , 1, 0, e, none.

Evaluate the integral: $\int \frac{d x}{x\left(\log _{8} x\right)^{2}}$ . Choose the correct answer from the options.

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Choices: $\frac{t}{\ln 2}$ , $\frac{\ln 8}{t}$ , none, $3+\ln t$ , $\frac{1}{t}$ .

Calculate the integral from 1 to 2 of $x^{x}(1+\ln (x))$ dx. What is the value? Options: 0, 3, 2, 1, none.

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list.

Evaluate the integral: $\int \frac{d x}{x \log _{10} x}$ . Select the correct answer from the options given.

Find the derivative $y'$ , where $y=x \sin(\log_{7} x)$ . Options include various expressions involving $\sin$ and $\cos$ .

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ . What is the value? Options: e, 2, 0, none, 1.

Calculate the integral $\int_{1}^{2} x^{x}(1+\ln (x)) d x$ .

Evaluate the integral $\int \frac{d x}{x(\log_{8} x)^{2}}$ and choose the correct answer.

Find the derivative of $y=x^{\ln (\ln (x))}$ at $x=e$ . What is the value? Options: $e$ , $e^{-1}$ , none, 0, 1.

Calculate the integral from 3 to 4 of $x^{x}(1+\ln (x))$ with respect to $x$ .

Find the derivative of $y=(x+1)^{x}$ . What is $y^{\prime}=$ ?

Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}+1}$ .

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{5 n}{n^{2}+1}$ using comparison tests.

Find the derivative of $y=x^{\ln (x)}$ for $x>0$ . Choose the correct option from the list provided.

Find the limit $\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}$ for the series $\sum_{n=1}^{\infty} \frac{(2 n) !}{(n !)(n+1) !}$ . Choices: a. 4 b. $\infty$ c. 0 d. 1 e. 2

Evaluate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choose from: 23, none, 1, 21, 5.

Find the derivative $y^{\prime}$ of $y=t \log _{3}(e^{(\sin t)(\ln 3)})$ .

Find the derivative of $y=(\ln x)^{\ln x}$ . What is $y'$ ?

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Choose: 21, none, 1, 23, 5.

Evaluate the integral from 2 to 3 of $x^{x}(1+\ln (x)) \, dx$ . What is the result? Options: 1, 5, 21, none, 23.

Evaluate the integral: $I = \int \frac{2 \log _{2}(x-1)}{(x-1)} d x$ . Choose the correct answer from the options.

Find the integral of $\frac{1}{x \log_{10} x}$ and choose the correct answer from the options given.

Find the derivative $y^{\prime}$ if $y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$ .

Find the derivative of $y=(\sqrt{t})^{t}$ . What is $y^{\prime}$ ?

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ . What is the result? Options: 23, none, 21, 5, 1.

Find the integral of $x^{2 x}(1+\ln x) \, dx$ . What is the result?

Find the derivative $y^{\prime}$ if $y=\log _{2}(8 t^{\ln 2})$ . Options: $3+\ln t$ , $\frac{1}{t}$ , $\frac{t}{\ln 2}$ , $\frac{\ln 8}{t}$ , none.

Find the derivative $y'$ , where $y = (x+1)^x$ . Choose from: $(x+1)^{x}(\frac{1}{x+1}+\ln(x+1))$ or others.

Find the derivative of $y=x \ln (\ln (x))$ at $x=e$ .

Calculate the integral from 2 to 3 of $x^{x}(1+\ln (x))$ dx. What is the result? Options: 21, 5, 23, 1.

Calculate the integral from $e^{2}$ to $e^{3}$ of $\frac{1+\ln (x)}{x \ln (x)} \, dx$ .

Find the derivative $y^{\prime}$ of $y=3 \log _{8}(\log _{2} t)$ . Choices include $\frac{3}{t(\ln t)(\ln 2)}$ and others.