CPM Homework Help : CALC Problem 12-51

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12-51.

Integrate each function.

Hint:

All of these integrals require the use of integration by parts.

$\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { x } d x$
Steps 1(a):
Note: The following steps ignore the bounds of integration. Be sure to apply them at the end.
Let $f = x^2$ and $dg = e^xdx$ .
Then $df = 2x dx$ and $g = e^x$ .
Steps 2(a):
For the integral:
Let $f = 2x$ and $dg = e^xdx$ .
Then $df = 2dx$ and $g = e^x$ .
Steps 3(a):
You can now integrate the integral that is left without using integration by parts.

$\int x ^ { 2 } e ^ { 3 x } d x$
Steps (b):
Use the same process outlined in part (a), but use $e^{3x}$ instead of $e^{x}$ .

$\int x \operatorname { sin } x d x$
Steps (c):
Let $f = x$ and $dg =\operatorname{sin}(x)$ .

$\int \operatorname { sin } ^ { 2 } x d x$
Steps (d):
Let $f =\operatorname{sin}(x)$ and $dg =\operatorname{sin}(x)dx$ .
Now solve this equation for the original integral.

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