Solved on Nov 17, 2023

Find the antiderivatives of $f(x)=-8 \sec^2 x$ and verify the solution by taking the derivative.

STEP 1

Assumptions1. The function we are working with is $f(x)=-8 \sec ^{} x$ . We are looking for all antiderivatives of this function3. We will check our work by taking the derivative of the antiderivative

STEP 2

To find the antiderivative of a function, we need to find a function whose derivative is the given function. In this case, we need to find a function whose derivative is $-8 \sec ^{2} x$ .

STEP 3

We know that the derivative of $\tan x$ is $\sec ^{2} x$ . Therefore, the antiderivative of $\sec ^{2} x$ is $\tan x$ .

STEP 4

Now, we can find the antiderivative of $-8 \sec ^{2} x$ by multiplying the antiderivative of $\sec ^{2} x$ by -8.
$(x) = -8 \tan x$

STEP 5

However, the antiderivative is not unique. There is a constant of integration, usually denoted by $C$ , that we need to add to the antiderivative. This is because the derivative of a constant is zero, so adding a constant to a function does not change its derivative.
$(x) = -8 \tan x + C$

STEP 6

Now, we will check our work by taking the derivative of $(x) = -8 \tan x + C$ .

STEP 7

The derivative of $\tan x$ is $\sec ^{2} x$ , and the derivative of a constant is zero. Therefore, the derivative of $- \tan x + C$ is $- \sec ^{2} x$ .
$f(x) = \frac{d}{dx}(- \tan x + C) = - \sec ^{2} x$

STEP 8

The derivative of our antiderivative is equal to the original function, so our work is correct.
The antiderivatives of $f(x)=-8 \sec ^{2} x$ are $(x)=-8 \tan x + C$ for all real numbers $C$ .

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Find the antiderivatives of $f(x)=-8 \sec^2 x$ and verify the solution by taking the derivative.

STEP 1

Assumptions1. The function we are working with is $f(x)=-8 \sec ^{} x$ . We are looking for all antiderivatives of this function3. We will check our work by taking the derivative of the antiderivative

STEP 2

To find the antiderivative of a function, we need to find a function whose derivative is the given function. In this case, we need to find a function whose derivative is $-8 \sec ^{2} x$ .

STEP 3

We know that the derivative of $\tan x$ is $\sec ^{2} x$ . Therefore, the antiderivative of $\sec ^{2} x$ is $\tan x$ .

STEP 4

Now, we can find the antiderivative of $-8 \sec ^{2} x$ by multiplying the antiderivative of $\sec ^{2} x$ by -8.
$(x) = -8 \tan x$

STEP 5

However, the antiderivative is not unique. There is a constant of integration, usually denoted by $C$ , that we need to add to the antiderivative. This is because the derivative of a constant is zero, so adding a constant to a function does not change its derivative.
$(x) = -8 \tan x + C$

STEP 6

Now, we will check our work by taking the derivative of $(x) = -8 \tan x + C$ .

STEP 7

The derivative of $\tan x$ is $\sec ^{2} x$ , and the derivative of a constant is zero. Therefore, the derivative of $- \tan x + C$ is $- \sec ^{2} x$ .
$f(x) = \frac{d}{dx}(- \tan x + C) = - \sec ^{2} x$

STEP 8

The derivative of our antiderivative is equal to the original function, so our work is correct.
The antiderivatives of $f(x)=-8 \sec ^{2} x$ are $(x)=-8 \tan x + C$ for all real numbers $C$ .

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Find the antiderivatives of f(x)=−8sec⁡2xf(x)=-8 \sec^2 xf(x)=−8sec2x and verify the solution by taking the derivative.

STEP 1

Assumptions1. The function we are working with is f(x)=−8sec⁡xf(x)=-8 \sec ^{} xf(x)=−8secx . We are looking for all antiderivatives of this function3. We will check our work by taking the derivative of the antiderivative

STEP 2

To find the antiderivative of a function, we need to find a function whose derivative is the given function. In this case, we need to find a function whose derivative is −8sec⁡2x-8 \sec ^{2} x−8sec2x.

STEP 3

We know that the derivative of tan⁡x\tan xtanx is sec⁡2x\sec ^{2} xsec2x. Therefore, the antiderivative of sec⁡2x\sec ^{2} xsec2x is tan⁡x\tan xtanx.

STEP 4

Now, we can find the antiderivative of −8sec⁡2x-8 \sec ^{2} x−8sec2x by multiplying the antiderivative of sec⁡2x\sec ^{2} xsec2x by -8.(x)=−8tan⁡x(x) = -8 \tan x(x)=−8tanx

STEP 5

STEP 6

Now, we will check our work by taking the derivative of (x)=−8tan⁡x+C(x) = -8 \tan x + C(x)=−8tanx+C.

STEP 7

STEP 8

The derivative of our antiderivative is equal to the original function, so our work is correct.The antiderivatives of f(x)=−8sec⁡2xf(x)=-8 \sec ^{2} xf(x)=−8sec2x are (x)=−8tan⁡x+C(x)=-8 \tan x + C(x)=−8tanx+C for all real numbers CCC.

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Find the antiderivatives of $f(x)=-8 \sec^2 x$ and verify the solution by taking the derivative.

Assumptions1. The function we are working with is $f(x)=-8 \sec ^{} x$ . We are looking for all antiderivatives of this function3. We will check our work by taking the derivative of the antiderivative

To find the antiderivative of a function, we need to find a function whose derivative is the given function. In this case, we need to find a function whose derivative is $-8 \sec ^{2} x$ .

We know that the derivative of $\tan x$ is $\sec ^{2} x$ . Therefore, the antiderivative of $\sec ^{2} x$ is $\tan x$ .

Now, we can find the antiderivative of $-8 \sec ^{2} x$ by multiplying the antiderivative of $\sec ^{2} x$ by -8.
$(x) = -8 \tan x$

Now, we will check our work by taking the derivative of $(x) = -8 \tan x + C$ .

The derivative of our antiderivative is equal to the original function, so our work is correct.
The antiderivatives of $f(x)=-8 \sec ^{2} x$ are $(x)=-8 \tan x + C$ for all real numbers $C$ .