  ### Home > APCALC > Chapter 12 > Lesson 12.1.4 > Problem12-46

12-46.

Integrate.

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

1. $\int _ { 0 } ^ { 1 } x ^ { 2 } e ^ { x } d x$

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 $g = e^{x}$.

$x^2e^x-\int2xe^xdx$

For the integral:
Let $f = 2x$ and $dg = e^{x}dx$.
Then $df = 2dx$ and $g = e^{x}$.

$x^2e^x-\Big(2xe^x-\int2e^xdx\Big)$

You can now integrate the integral that is left without using integration by parts.

1. $\int x ^ { 2 } e ^ { 3 x } d x$

Use the same process outlined in part (a), but use $e^{3x}$ instead of $e^{x}$.

1. $\int x \operatorname { sin } ( x ) d x$

Let $f = x$ and $dg = \sin\left(x\right)$.

1. $\int \operatorname { sin } ^ { 2 } ( x ) d x$

Let $f = \sin\left(x\right)$ and $dg = \sin\left(x\right)dx$.

$-\sin(x)\cos(x)+\int\cos^2(x)dx$

$=-\sin(x)\cos(x)+\int(1-\sin^2(x))dx$

$\int\sin^2(x)dx=-\sin(x)\cos(x)+x-\int\sin^2(x)dx$

Now solve this equation for the original integral.