### Home > CALC3RD > Chapter Ch7 > Lesson 7.3.6 > Problem7-166

7-166.

Evaluate each integral below. Show your steps. If you use $u$-substitution, be sure to change the bounds of integration.

1. $\int _ { \pi / 4 } ^ { 0 } \sqrt { \operatorname { sin } ( x ) } \cdot \operatorname { cos } ( x ) d x$

$U = \sin(x)$
$dU = \cos(x)dx$

$U\left ( \frac{\pi }{4} \right )=\text{ lower bound}$

$U(0) = \text{ upper bound}$

Assemble the integral and then solve.

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

$U = \sin(x)$
$dU = \cos(x)dx$

1. $\int \frac { 3 x ^ { 4 } - 4 x ^ { 2 } - 11 x + 6 } { x ^ { 2 } } d x$

Before you integrate, rewrite the integrand.

$= \int 3x^{2}-4-\frac{11}{x}+\frac{6}{x^{2}}dx=$

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

$\sec^{−1}(x) + C$ (this is a special case)

1. $\int _ { 1 } ^ { e } \frac { 1 } { x } \operatorname { cos } ( \operatorname { ln } ( x ) ) d x$

$\frac{d}{dx}\left ( \frac{1}{x} \right )=\ln(x)$

1. $\int _ { - 1 } ^ { 3 } x ^ { 2 } ( x ^ { 3 } - 8 ) d x$

You could use $U$-substitution.
Or, you could expand the integrand and evaluate.