CPM Homework Help : APCALC Problem 10-4

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10-4.

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.

$\int x ^ { 2 } e ^ { - x } d x$
Hint (a):
Use integration by parts.
Let $f = x^2$ and $dg = e^{–x}$ .

$\int x \operatorname { sec } ( x ^ { 2 } ) d x$
Hint (b):
Use substitution.
Let $u = x^2$ .
More Help:
$\int\sec(x)dx=\ln|\sec(x)+\tan(x)|$

$\int _ { - \infty } ^ { \infty } \frac { 1 } { 1 + x ^ { 2 } } d x$
Hint (c):
You should recognize the integrand as the derivative of an inverse trigonometric function.
More Help:
This is an improper integral, so a limit will need to be used to evaluate it.
The symmetry of the graph can also be used to evaluate the integral.

$\int _ { a } ^ { b } f ^ { \prime } ( x ) d x$
Hint (d):
Review the first Fundamental Theorem of Calculus.

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