Question

Need to solve using integral by parts
13x^2 sin (pi x) dx

Answer 1

\[Let I=\int\limits 13 x^{2}\sin \left( \pi x \right) dx=13\left[ x ^{2}\left\{ \frac{-\cos \pi x }{ \pi } \right\}-\int\limits 2x \left\{ \frac{ -\cos \pi x }{\pi } \right\} dx\right]\]
\[=13\left[ \frac{ -x ^{2}\cos \pi x }{ \pi }+\frac{ 2 }{ \pi }\left\{ x \frac{ \sin \pi x }{ \pi }-\int\limits 1\frac{ \sin \pi x }{ \pi }dx \right\} \right]\]
\[=13\left[ \frac{ -x ^{2}\cos \pi x }{ \pi }+\frac{ 2x \sin \pi x }{ \pi ^{2}}+\frac{ 2\cos \pi x }{\pi ^{3} } \right]+c\]

Answer 2

\[\int\limits uvdx=u \int\limits vdx-\int\limits u \prime( \int\limits v dx ) dx\]