Solved on Jan 17, 2024

Find the most general antiderivative of the function $f(x) = 3e^x + 8\sec^2 x$ . Answer: $F(x) = 3e^x - 8\tan x + C$ .

STEP 1

Assumptions
1. The function to differentiate is $f(x) = 3e^x + 8\sec^2x$ .
2. We are looking for the most general antiderivative, which means we will include an arbitrary constant $C$ in our final answer.

STEP 2

The antiderivative of a sum of functions is the sum of the antiderivatives of the individual functions. Therefore, we can find the antiderivative of $f(x)$ by finding the antiderivatives of $3e^x$ and $8\sec^2x$ separately.

STEP 3

First, let's find the antiderivative of $3e^x$ . The antiderivative of $e^x$ is itself, so the antiderivative of $3e^x$ is simply $3e^x$ .

STEP 4

Now, let's find the antiderivative of $8\sec^2x$ . Recall that the derivative of $\tan(x)$ is $\sec^2(x)$ , so the antiderivative of $\sec^2(x)$ is $\tan(x)$ .

STEP 5

Multiply the antiderivative of $\sec^2(x)$ by 8 to get the antiderivative of $8\sec^2x$ , which is $8\tan(x)$ .

STEP 6

Combine the antiderivatives from steps 3 and 5 to get the antiderivative of the original function $f(x)$ .
$F(x) = 3e^x + 8\tan(x)$

STEP 7

Since we are looking for the most general antiderivative, we must add an arbitrary constant $C$ to our result.

STEP 8

The most general antiderivative of the function $f(x)$ is:
$F(x) = 3e^x + 8\tan(x) + C$
This is the final answer.

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Find the most general antiderivative of the function $f(x) = 3e^x + 8\sec^2 x$ . Answer: $F(x) = 3e^x - 8\tan x + C$ .

STEP 1

Assumptions
1. The function to differentiate is $f(x) = 3e^x + 8\sec^2x$ .
2. We are looking for the most general antiderivative, which means we will include an arbitrary constant $C$ in our final answer.

STEP 2

The antiderivative of a sum of functions is the sum of the antiderivatives of the individual functions. Therefore, we can find the antiderivative of $f(x)$ by finding the antiderivatives of $3e^x$ and $8\sec^2x$ separately.

STEP 3

First, let's find the antiderivative of $3e^x$ . The antiderivative of $e^x$ is itself, so the antiderivative of $3e^x$ is simply $3e^x$ .

STEP 4

Now, let's find the antiderivative of $8\sec^2x$ . Recall that the derivative of $\tan(x)$ is $\sec^2(x)$ , so the antiderivative of $\sec^2(x)$ is $\tan(x)$ .

STEP 5

Multiply the antiderivative of $\sec^2(x)$ by 8 to get the antiderivative of $8\sec^2x$ , which is $8\tan(x)$ .

STEP 6

Combine the antiderivatives from steps 3 and 5 to get the antiderivative of the original function $f(x)$ .
$F(x) = 3e^x + 8\tan(x)$

STEP 7

Since we are looking for the most general antiderivative, we must add an arbitrary constant $C$ to our result.

STEP 8

The most general antiderivative of the function $f(x)$ is:
$F(x) = 3e^x + 8\tan(x) + C$
This is the final answer.

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Find the most general antiderivative of the function f(x)=3ex+8sec⁡2xf(x) = 3e^x + 8\sec^2 xf(x)=3ex+8sec2x. Answer: F(x)=3ex−8tan⁡x+CF(x) = 3e^x - 8\tan x + CF(x)=3ex−8tanx+C.

STEP 1

Assumptions1. The function to differentiate is f(x)=3ex+8sec⁡2x f(x) = 3e^x + 8\sec^2x f(x)=3ex+8sec2x.2. We are looking for the most general antiderivative, which means we will include an arbitrary constant C C C in our final answer.

STEP 2

The antiderivative of a sum of functions is the sum of the antiderivatives of the individual functions. Therefore, we can find the antiderivative of f(x) f(x) f(x) by finding the antiderivatives of 3ex 3e^x 3ex and 8sec⁡2x 8\sec^2x 8sec2x separately.

STEP 3

First, let's find the antiderivative of 3ex 3e^x 3ex. The antiderivative of ex e^x ex is itself, so the antiderivative of 3ex 3e^x 3ex is simply 3ex 3e^x 3ex.

STEP 4

Now, let's find the antiderivative of 8sec⁡2x 8\sec^2x 8sec2x. Recall that the derivative of tan⁡(x) \tan(x) tan(x) is sec⁡2(x) \sec^2(x) sec2(x), so the antiderivative of sec⁡2(x) \sec^2(x) sec2(x) is tan⁡(x) \tan(x) tan(x).

STEP 5

Multiply the antiderivative of sec⁡2(x) \sec^2(x) sec2(x) by 8 to get the antiderivative of 8sec⁡2x 8\sec^2x 8sec2x, which is 8tan⁡(x) 8\tan(x) 8tan(x).

STEP 6

Combine the antiderivatives from steps 3 and 5 to get the antiderivative of the original function f(x) f(x) f(x).F(x)=3ex+8tan⁡(x) F(x) = 3e^x + 8\tan(x) F(x)=3ex+8tan(x)

STEP 7

Since we are looking for the most general antiderivative, we must add an arbitrary constant C C C to our result.

STEP 8

The most general antiderivative of the function f(x) f(x) f(x) is:F(x)=3ex+8tan⁡(x)+C F(x) = 3e^x + 8\tan(x) + C F(x)=3ex+8tan(x)+CThis is the final answer.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the most general antiderivative of the function $f(x) = 3e^x + 8\sec^2 x$ . Answer: $F(x) = 3e^x - 8\tan x + C$ .

Assumptions
1. The function to differentiate is $f(x) = 3e^x + 8\sec^2x$ .
2. We are looking for the most general antiderivative, which means we will include an arbitrary constant $C$ in our final answer.

The antiderivative of a sum of functions is the sum of the antiderivatives of the individual functions. Therefore, we can find the antiderivative of $f(x)$ by finding the antiderivatives of $3e^x$ and $8\sec^2x$ separately.

First, let's find the antiderivative of $3e^x$ . The antiderivative of $e^x$ is itself, so the antiderivative of $3e^x$ is simply $3e^x$ .

Now, let's find the antiderivative of $8\sec^2x$ . Recall that the derivative of $\tan(x)$ is $\sec^2(x)$ , so the antiderivative of $\sec^2(x)$ is $\tan(x)$ .

Multiply the antiderivative of $\sec^2(x)$ by 8 to get the antiderivative of $8\sec^2x$ , which is $8\tan(x)$ .

Combine the antiderivatives from steps 3 and 5 to get the antiderivative of the original function $f(x)$ .
$F(x) = 3e^x + 8\tan(x)$

Since we are looking for the most general antiderivative, we must add an arbitrary constant $C$ to our result.

The most general antiderivative of the function $f(x)$ is:
$F(x) = 3e^x + 8\tan(x) + C$
This is the final answer.