Integration By Parts help

In summary, the conversation discusses two integrals: $\int x^ne^xdx$ and $\int \sin ^nxdx$. The first one can be solved using integration by parts, while the second one depends on whether the power of sine is odd or even.
  • #1
Slimsta
190
0

Homework Statement


1.[tex]$\int x^ne^xdx$[/tex]
2.[tex]$\int \sin ^nxdx$[/tex]


Homework Equations


[tex]$ \displaystyle \Large \int fg dx = fg - \int gf' dx$ [/tex]


The Attempt at a Solution


1. f=xn
g'=ex
g=ex
f'=nxn-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) ?
 
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  • #2
No, the anti-derivative of sinn(x) is NOT sinn+1(x)/(n+1)!

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

[tex]\int x^n e^x dx= x^ne^x- n\int x^{n-1}e^x dx[/tex]
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
[tex]\int sin^{2m+1}(x) dx= \int (sin^2(x))^m sin(x)dx= \int (1- cos^2(x))^msin(x) dx[/tex]
and letting u= cos(x) reduces it to
[tex]-\int (1- u^2)^m du[/tex]

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

What is integration by parts?

Integration by parts is a technique used in calculus to simplify the integration of a product of two functions.

When should integration by parts be used?

Integration by parts should be used when the integrand is a product of two functions and cannot be easily simplified by substitution or other methods.

How do you use integration by parts?

To use integration by parts, the integrand is split into two separate functions, one to be differentiated and one to be integrated. The resulting terms are then substituted into the integration by parts formula and solved for the integral.

What is the integration by parts formula?

The integration by parts formula is ∫u dv = uv - ∫v du, where u and v are the two functions and du and dv are their respective differentials.

What are some tips for using integration by parts?

Some tips for using integration by parts include choosing u to be the more complicated function, choosing dv to be easily integrable, and using integration by parts multiple times if necessary.

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