Solved on Jan 17, 2024

Find the most general antiderivative of the function f(x)=3ex+8sec2xf(x) = 3e^x + 8\sec^2 x. Answer: F(x)=3ex8tanx+CF(x) = 3e^x - 8\tan x + C.

STEP 1

Assumptions
1. The function to differentiate is f(x)=3ex+8sec2x 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 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) by finding the antiderivatives of 3ex 3e^x and 8sec2x 8\sec^2x separately.

STEP 3

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

STEP 4

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

STEP 5

Multiply the antiderivative of sec2(x) \sec^2(x) by 8 to get the antiderivative of 8sec2x 8\sec^2x , which is 8tan(x) 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) .
F(x)=3ex+8tan(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 C to our result.

STEP 8

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

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