Solved on Nov 20, 2023

Integrate $(t+7)$ to find $F(x)$ , then demonstrate the Second Fundamental Theorem by differentiating $F(x)$ .

STEP 1

Assumptions1. The function to be integrated is $f(t) = t +7$ . . The limits of integration are from0 to x.
3. We are asked to find the antiderivative of $f(t)$ with respect to $t$ , and then differentiate the result to demonstrate the Second Fundamental Theorem of Calculus.

STEP 2

First, we need to find the antiderivative of $f(t) = t +7$ . The antiderivative of a function is the function whose derivative is the original function.
$(x) = \int_{0}^{x} (t +7) dt$

STEP 3

The integral of a sum of functions is the sum of the integrals of the functions. Therefore, we can rewrite the integral as $(x) = \int_{0}^{x} t dt + \int_{0}^{x}7 dt$

STEP 4

Now, we can integrate each term separately. The antiderivative of $t$ with respect to $t$ is $\frac{1}{2}t^2$ and the antiderivative of $7$ with respect to $t$ is $7t$ .
$(x) = \left[\frac{1}{2}t^2\right]_{0}^{x} + \left[7t\right]_{0}^{x}$

STEP 5

Now, we can evaluate each term at the upper and lower limits of integration. $(x) = \frac{1}{2}x^2 - \frac{1}{2}(0)^2 +7x -7(0)$

STEP 6

implify the expression to get $(x)$ .
$(x) = \frac{1}{2}x^2 +x$

STEP 7

Now, we need to differentiate $(x)$ to demonstrate the Second Fundamental Theorem of Calculus. The theorem states that if a function $f$ is continuous over the interval $[a, b]$ and $$ is an antiderivative of $f$ on $[a, b]$, then the derivative of the integral of $f$ from $a$ to $x$ is $f(x)$.
$'(x) = \frac{d}{dx} \left(\frac{1}{2}x^2 +7x\right)$

STEP 8

Apply the power rule and the constant rule to differentiate each term. The derivative of $x^n$ is $nx^{n-1}$ and the derivative of a constant times $x$ is the constant.
$'(x) = x +7$ So, $(x) = \frac{1}{2}x^2 +7x$ and $'(x) = x +7$ .

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Integrate $(t+7)$ to find $F(x)$ , then demonstrate the Second Fundamental Theorem by differentiating $F(x)$ .

STEP 1

Assumptions1. The function to be integrated is $f(t) = t +7$ . . The limits of integration are from0 to x.
3. We are asked to find the antiderivative of $f(t)$ with respect to $t$ , and then differentiate the result to demonstrate the Second Fundamental Theorem of Calculus.

STEP 2

First, we need to find the antiderivative of $f(t) = t +7$ . The antiderivative of a function is the function whose derivative is the original function.
$(x) = \int_{0}^{x} (t +7) dt$

STEP 3

The integral of a sum of functions is the sum of the integrals of the functions. Therefore, we can rewrite the integral as $(x) = \int_{0}^{x} t dt + \int_{0}^{x}7 dt$

STEP 4

Now, we can integrate each term separately. The antiderivative of $t$ with respect to $t$ is $\frac{1}{2}t^2$ and the antiderivative of $7$ with respect to $t$ is $7t$ .
$(x) = \left[\frac{1}{2}t^2\right]_{0}^{x} + \left[7t\right]_{0}^{x}$

STEP 5

Now, we can evaluate each term at the upper and lower limits of integration. $(x) = \frac{1}{2}x^2 - \frac{1}{2}(0)^2 +7x -7(0)$

STEP 6

implify the expression to get $(x)$ .
$(x) = \frac{1}{2}x^2 +x$

STEP 7

STEP 8

Apply the power rule and the constant rule to differentiate each term. The derivative of $x^n$ is $nx^{n-1}$ and the derivative of a constant times $x$ is the constant.
$'(x) = x +7$ So, $(x) = \frac{1}{2}x^2 +7x$ and $'(x) = x +7$ .

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Integrate (t+7)(t+7)(t+7) to find F(x)F(x)F(x), then demonstrate the Second Fundamental Theorem by differentiating F(x)F(x)F(x).

STEP 1

Assumptions1. The function to be integrated is f(t)=t+7f(t) = t +7f(t)=t+7. . The limits of integration are from0 to x.3. We are asked to find the antiderivative of f(t)f(t)f(t) with respect to ttt, and then differentiate the result to demonstrate the Second Fundamental Theorem of Calculus.

STEP 2

First, we need to find the antiderivative of f(t)=t+7f(t) = t +7f(t)=t+7. The antiderivative of a function is the function whose derivative is the original function.(x)=∫0x(t+7)dt(x) = \int_{0}^{x} (t +7) dt(x)=∫0x​(t+7)dt

STEP 3

The integral of a sum of functions is the sum of the integrals of the functions. Therefore, we can rewrite the integral as(x)=∫0xtdt+∫0x7dt(x) = \int_{0}^{x} t dt + \int_{0}^{x}7 dt(x)=∫0x​tdt+∫0x​7dt

STEP 4

Now, we can integrate each term separately. The antiderivative of ttt with respect to ttt is 12t2\frac{1}{2}t^221​t2 and the antiderivative of 777 with respect to ttt is 7t7t7t.(x)=[12t2]0x+[7t]0x(x) = \left[\frac{1}{2}t^2\right]_{0}^{x} + \left[7t\right]_{0}^{x}(x)=[21​t2]0x​+[7t]0x​

STEP 5

Now, we can evaluate each term at the upper and lower limits of integration.(x)=12x2−12(0)2+7x−7(0)(x) = \frac{1}{2}x^2 - \frac{1}{2}(0)^2 +7x -7(0)(x)=21​x2−21​(0)2+7x−7(0)

STEP 6

implify the expression to get (x)(x)(x).(x)=12x2+x(x) = \frac{1}{2}x^2 +x(x)=21​x2+x

STEP 7

STEP 8

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Integrate $(t+7)$ to find $F(x)$ , then demonstrate the Second Fundamental Theorem by differentiating $F(x)$ .

Assumptions1. The function to be integrated is $f(t) = t +7$ . . The limits of integration are from0 to x.
3. We are asked to find the antiderivative of $f(t)$ with respect to $t$ , and then differentiate the result to demonstrate the Second Fundamental Theorem of Calculus.

First, we need to find the antiderivative of $f(t) = t +7$ . The antiderivative of a function is the function whose derivative is the original function.
$(x) = \int_{0}^{x} (t +7) dt$

The integral of a sum of functions is the sum of the integrals of the functions. Therefore, we can rewrite the integral as $(x) = \int_{0}^{x} t dt + \int_{0}^{x}7 dt$

Now, we can integrate each term separately. The antiderivative of $t$ with respect to $t$ is $\frac{1}{2}t^2$ and the antiderivative of $7$ with respect to $t$ is $7t$ .
$(x) = \left[\frac{1}{2}t^2\right]_{0}^{x} + \left[7t\right]_{0}^{x}$

Now, we can evaluate each term at the upper and lower limits of integration. $(x) = \frac{1}{2}x^2 - \frac{1}{2}(0)^2 +7x -7(0)$

implify the expression to get $(x)$ .
$(x) = \frac{1}{2}x^2 +x$