CPM Homework Help : CALC Problem 11-97

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11-97.

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.

$\int _ { 0 } ^ { 2 } \frac { 1 } { ( x - 2 ) ^ { 2 } } d x$
Hint (a):
The Reverse Chain Rule can be used here, but this is an improper integral, so be sure to use a limit.

$\int \operatorname { sec } ^ { 2 } x \operatorname { ln } ( \operatorname { tan } x ) d x$
Step (b):
Let $u =\operatorname{tan}(x)$ .

$\int \frac { 3 } { ( x - 1 ) ( x + 2 ) } d x$
Hint (c):
Use partial fraction decomposition.
$\frac{A}{x-1}+\frac{B}{x+2}=\frac{3}{(x-1)(x+2)}$

$\int 6 x \operatorname { tan } x ^ { 2 } d x$
Hint (d):
Use substitution twice.
Step 1(d):
Let $u = x^2$ . Then $du = 2x$ .
$= \int 3\tan(u)du$
Step 2(d):
Let $v =\operatorname{cos}(u)$ . Then $dv = –\operatorname{sin}(u)du$ .
$=-3\int\frac{1}{v}dv$

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