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10-80.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy. After evaluating each integral, write a short description of your method.

$\int \frac { \operatorname { sec } ^ { 2 } t } { 1 + \operatorname { tan } t } d t$
Hint (a):
Use substitution. Let $u =\operatorname{tan}(t)$ .

$\int _ { 0 } ^ { 2 } \frac { 1 } { ( x - 2 ) ( x + 1 ) } d x$
Hint (b):
Use partial fraction decomposition.
More Help (b):
$1/(x – 2)$ is undefined when $x = 2$ , so a limit needs to be used to evaluate this integral.

$\int \frac { d y } { d x } ( e ^ { \sqrt { 2 x } } ) d x$
Hint (c):
Since this is the integral of a derivative, your answer should have $a^{\prime\prime}+ C^{\prime\prime}$ .

$\int \frac { 1 } { x \sqrt { \operatorname { ln } x } } d x$
Hint (d):
Use substitution. Let $u =\operatorname{ln}(x)$ .

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