CPM Homework Help : CALC3RD Problem 10-136

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10-136.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method.

$\int \operatorname { sec } ^ { 4 } ( x ) \operatorname { tan } ^ { 3 } ( x ) d x$
Hint (a):
Rewrite the integrand in terms of sine and cosine and use the Pythagorean Identity.
Step (a):
$=\int\frac{(1-\cos^2(x))\sin(x)}{\cos^7(x)}dx$
More Help (a):
Let $u = cos(x)$ .

$\int x \operatorname { tan } ^ { - 1 } ( x ) d x$
Step 1 (b):
Use integration by parts: Let $f=\tan^{-1}\left(x\right)$ and $dg = x dx$
$df=\frac{1}{1+x^2}\text{, }g=\frac{1}{2}x^2$
Step 2 (b):
$\int x\tan^{-1}(x)dx=\frac{1}{2}x^2\tan^{-1}(x)-\frac{1}{2}\int \frac{x^2}{1+x^2}dx$
Step 3 (b):
Use polynomial division: $=\frac{1}{2}x^2\tan^{-1}(x)-\frac{1}{2}\int \Big(1-\frac{1}{1+x^2}\Big)dx$

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

$\int \frac { \operatorname { cos } \sqrt { x } } { \sqrt { x } } d x$
Hint (d):
Use substitution:
$\text{Let }u=\sqrt{x}$

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