Cracking The Ap Calculus ab Exam 2018

y = ∫ (3x + 5) dx

Integrating, we get

y   =       +   5x  +   C

Now we can solve for the constant because we know that y = 6 when x = 0.

6 = + 5(0) + C

Therefore, C = 6 and the equation is

y   =       +   5x  +   6

Example 13: Find f(x) if f′(x) = sin x − cos x and f(π) = 3.

Integrate f′(x).

f(x) = ∫ (sin x − cos x) dx = −cos x − sin x + C

Now solve for the constant.

3   =   −cos(π) −   sin(π)  +   C

C   =   2

Therefore, the equation becomes

f(x)    =   −cos    x   −   sin x   +   2

Now we’ve covered the basics of integration. However, integration is a very sophisticated topic and
there are many types of integrals that will cause you trouble. We will need several techniques to learn
how to evaluate these integrals. The first and most important is called u-substitution, which we will cover
in the second half of this chapter.

In the meantime, here are some solved problems. Do each problem, covering the answer first, then check
your answer.

PROBLEM 1. Evaluate

Answer: Here’s the Power Rule again.