3. -/6.25 points details my notes evaluate the integral using integration by parts with the indicated…

Answer:

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

Explanation:

Step1: Find $du$ and $v$

$u = x$, so $du=dx$; $dv=\cos(5x)dx$, integrating gives $v=\frac{1}{5}\sin(5x)$

Step2: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Substituting $u$, $v$, $du$ into the formula: $\int x\cos(5x)dx=x\cdot\frac{1}{5}\sin(5x)-\int\frac{1}{5}\sin(5x)dx$

Step3: Integrate the remaining integral

$\int\frac{1}{5}\sin(5x)dx=-\frac{1}{25}\cos(5x)+C_1$ So $\int x\cos(5x)dx=\frac{1}{5}x\sin(5x)+\frac{1}{25}\cos(5x)+C$ (where $C = C_1$)