1. An Intergral Check

$\displaystyle \int x^n sin(x) * dx$ = ______________________ + $\displaystyle \int x^{n-1} cos(x) * dx$

I want to just say sin(x), but that n should be a +1 and not minus or am I wrong? And,

$\displaystyle Evaluate \int 5x * sin(x) * dx$

Would you just pull the 5 out and call it a day or what?

2. Originally Posted by Selim
$\displaystyle \int x^n sin(x) * dx$ = ______________________ + $\displaystyle \int x^{n-1} cos(x) * dx$

I want to just say sin(x), but that n should be a +1 and not minus or am I wrong? And,

$\displaystyle Evaluate \int 5x * sin(x) * dx$

Would you just pull the 5 out and call it a day or what?
in the blank cos(x) will come as it is the integration of sin(x). in ur second question as u said simply pull out 5 and integrate by parts.

3. Originally Posted by Selim
$\displaystyle \int x^n sin(x) * dx$ = ______________________ + $\displaystyle \int x^{n-1} cos(x) * dx$

I want to just say sin(x), but that n should be a +1 and not minus or am I wrong? And,

$\displaystyle Evaluate \int 5x * sin(x) * dx$

Would you just pull the 5 out and call it a day or what?
Using integration by parts

Let $\displaystyle u = x^n$ so that $\displaystyle du = nx^{n - 1}$

Let $\displaystyle dv = \sin{x}$ so that $\displaystyle v = -\cos{x}$.

Then $\displaystyle \int{x^n\sin{x}\,dx} = -x^n\cos{x} - \int{-nx^{n - 1}\cos{x}\,dx}$

$\displaystyle = -x^n\cos{x} + n\int{x^{n - 1}\cos{x}\,dx}$.

4. never mind