CPM Homework Help : CALC Problem 11-22

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11-22.

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.

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

$\int \frac { \operatorname { cos } ^ { 2 } x - \operatorname { sin } ^ { 2 } x } { \operatorname { sin } 2 x } d x$
Hint (b):
$\operatorname{cos}^2(x)-\operatorname{sin}^2(x) =\operatorname{cos}(2x)$
More Help (b):
Use $u$ -substitution. Let $u =\operatorname{sin}(2x)$ .

$\int _ { 0 } ^ { a } ( a x ^ { 2 / 3 } - b ) d x$
Step (c):
Note: Given $dx$ , $x$ is the only variable.
$\left.=\frac{3}{5}ax^{5/3}-bx\right|_0^a$

$\int \frac { 1 } { 2 x \sqrt { x ^ { 2 } - 1 } } d x$
Hint (d):
The integrand is a multiple of an inverse trigonometric function. Which one?

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