How Do You Solve Integrals Using Integration By Parts?

[Slimsta]

Homework Statement

1.$$\int x^ne^xdx$$
2.$$\int \sin ^nxdx$$

Homework Equations

$$\displaystyle \Large \int fg dx = fg - \int gf' dx$$

The Attempt at a Solution

1. f=xⁿ
g'=e^x
g=e^x
f'=nx^n-1

then just plug it in the formula? i tried but i don't get the right answer..

2. i have no idea how to even start..
the antiderivative of sinⁿx is [(sinx)ⁿ⁺¹]/(n+1) ?

 

[HallsofIvy]

No, the anti-derivative of sinⁿ(x) is NOT sinⁿ⁺¹(x)/(n+1)!

Integrating $\int x^n e^x dx$ is very easy- but tedious. Let u= xⁿ, dv= e^x dx. Then du= n x^n-1 dx and v= e^x.

$$\int x^n e^x dx= x^ne^x- n\int x^{n-1}e^x dx$$
That is the same as you started with but the exponent on x is one less. Repeat n-1 more times until the exponent is 0!

As for (2), how you do that depends on whether n is even or odd. If n is odd, it is easy. If n= 2m+1, then the integral is
$$\int sin^{2m+1}(x) dx= \int (sin^2(x))^m sin(x)dx= \int (1- cos^2(x))^msin(x) dx$$
and letting u= cos(x) reduces it to
$$-\int (1- u^2)^m du$$

If n is even, use the trig identity $sin^2(x)= (1/2)(1- cos(2x))$ repeatedly until you have reduced to power 1.