Calc1.59

From Example Problems
Jump to: navigation, search

\int (x^{3}+1)\cos x\,dx\,

Since we have x^{3}+1\, multiplied by \cos x\,, we know that it will take 3 applications of integration by parts to complete this problem.

u=x^{3}+1\,

du=3x^{2}\,dx\,

dv=\cos x\,dx\,

v=\sin x\,

\int (x^{3}+1)\cos x\,dx=(x^{3}+1)\sin x-3\int x^{2}\sin x\,dx\,

u=x^{2}\,

du=2x\,dx\,

dv=\sin x\,dx\,

v=-\cos x\,

\int (x^{3}+1)\cos x\,dx=(x^{3}+1)\sin x-3\left[-x^{2}\cos x+2\int x\cos x\,dx\right]\,

u=x\,

du=dx\,

dv=\cos x\,dx\,

v=\sin x\,

\int (x^{3}+1)\cos x\,dx=(x^{3}+1)\sin x+3x^{2}\cos x-6\left[x\sin x-\int \sin x\,dx\right]\,

=(x^{3}+1)\sin x+3x^{2}\cos x-6x\sin x-6\cos x+C\,


Main Page : Calculus