Cracking The Ap Calculus ab Exam 2018

You’ve learned how to integrate polynomials and some of the trig functions (there are more of them to
come), and you have the first technique of integration: u-substitution. Now it’s time to learn how to
integrate some other functions—namely, exponential and logarithmic functions. Yes, the long-awaited
second part of Chapter 11. The first integral is the natural logarithm.

    =   ln  |u| +   C

Notice the absolute value in the logarithm. This ensures that you aren’t taking the logarithm of a negative

number. If you know that the term you’re taking the log of is positive (for example, x^2 + 1), we can
dispense with the absolute value marks. Let’s do some examples.

Example 8: Find

Whenever an integrand contains a fraction, check to see if the integral is a logarithm. Usually, the process
involves u-substitution. Let u = x + 3 and du = dx. Then,

Substituting back, the final result is

5   ln  |x  +   3|  +   C

Example 9: Find

Let u = x^2 + 1 du = 2x dx and substitute into the integrand.

Then substitute back.

ln  (x^2    +   1)  +   C

MORE INTEGRALS OF TRIG FUNCTIONS

Remember when we started antiderivatives and we didn’t do the integral of tangent, cotangent, secant, or