### Home > APCALC > Chapter 11 > Lesson 11.1.2 > Problem11-22

11-22.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method.

1. $\int \frac { 1 } { 2 } \operatorname { sec } ^ { 2 } ( x ) \operatorname { tan } ( x ) d x$

Use $u$-substitution. Let $u = \sec(x)$.

1. $\int \frac { \operatorname { cos } ^ { 2 } ( x ) - \operatorname { sin } ^ { 2 } ( x ) } { \operatorname { sin } ( 2 x ) } d x$

$\cos^2(x) - \sin^2(x) = \cos(2x)$

Use $u$-substitution. Let $u = \sin(2x)$.

1. $\int _ { 0 } ^ { a } ( a x ^ { 2 / 3 } - b ) d x$

Note: Given $dx$, $x$ is the only variable.

$\left.=\frac{3}{5}ax^{5/3}-bx\right|_0^a$

1. $\int \frac { 1 } { 2 x \sqrt { x ^ { 2 } - 1 } } d x$

The integrand is a multiple of an inverse trigonometric function. Which one?