evaluate the integral using integration by parts.\n int xsin(5x)dx \n int xsin(5x)dx=square

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = x$ and $\mathrm{d}v=\sin(5x)\mathrm{d}x$. Then $\mathrm{d}u=\mathrm{d}x$. To find $v$, integrate $\mathrm{d}v$: $\int\sin(5x)\mathrm{d}x=-\frac{1}{5}\cos(5x)+C$, so $v = -\frac{1}{5}\cos(5x)$.

Step2: Substitute into the formula

$\int x\sin(5x)\mathrm{d}x=u v-\int v\mathrm{d}u=x\left(-\frac{1}{5}\cos(5x)\right)-\int\left(-\frac{1}{5}\cos(5x)\right)\mathrm{d}x$.

Step3: Evaluate the remaining integral

$\int\left(-\frac{1}{5}\cos(5x)\right)\mathrm{d}x=-\frac{1}{25}\sin(5x)+C$. So $\int x\sin(5x)\mathrm{d}x=-\frac{1}{5}x\cos(5x)+\frac{1}{25}\sin(5x)+C$.

Answer:

$-\frac{1}{5}x\cos(5x)+\frac{1}{25}\sin(5x)+C$