AP Calculus BC Question 87: Answer and Explanation

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Question: 87

5. sec4 x dx =

  • A. tan4 x + C
  • B. tan x + tan3 x + C
  • C. tan2 x + C
  • D. + C

Correct Answer: B


B First, break up the integrand: sec4 x dx = (sec2 x)(sec2 x) dx.

Next, use the trig identity 1 + tan2 x = sec2 x to rewrite the integral.

We can evaluate these integrals separately.

The left one is easy: sec2 x dx = tan x + C.

We will use u-substitution for the right one. Let u = tan x and du = sec2 x dx. Then, substitute into the integral and integrate: .

Now, substitute back: tan3 x + C.

Combine the two integrals to get tan x + tan3 x + C.