### Home > APCALC > Chapter 8 > Lesson 8.1.4 > Problem8-42

8-42.

No calculator! Evaluate the integrals below.

1. $\int _ { 2 } ^ { 4 } \frac { 1 } { 9 - 2 x } d x$

Before integrating, make a choice:
Factor out $-1/2$ or let $u = 9 - 2x$

$\int \frac{1}{x}dx=\text{ln}\left | x \right |+C$

Don't forget the absolute value!

$\text{ln}|a|-\text{ln}|b|=\text{ln}\left | \frac{a}{b} \right |$

1. $\int _ { - 3 } ^ { 3 } ( 2 x ^ { 5 } + 3 x ^ { 2 } + 1 ) d x$

Notice the bounds. Is this function odd? If so, the area will be $0$. If not, integrate.

1. $\int _ { 3 } ^ { 2 } \frac { x ^ { 2 } - 1 } { x - 1 } d x$

Factor before integrating.

1. $\int _ { 1 } ^ { 4 } \sqrt { 16 x ^ { 3 } } d x$

$\sqrt{16x^{3}}=4x^{3/2}$

1. $\int _ { \pi / 4 } ^ { \pi / 2 } \operatorname { sin } ^ { 3 } ( x ) \operatorname { cos } ( x ) d x$

Use $u$-substitution. Should $u = \sin(x)$ or$u = \cos(x)$?