Question

Evaluate the integral:-

i)\(x\cos 5xdx\)

Short answer

We shall use the integration by parts method to solve the given integral.

Let u=x and dv=cos5xdx

Evaluate du and v and substitute them in

\(\int {x\cos 5x = } \frac{x}{5}\sin 5x + \frac{{\cos 5x}}{{25}} + c\)\(\int {udv = uv - \int {vdu} } \)

Step-by-step solution

Step 1- Given Data

Given:- I=\(\int {x\cos 5xdx} \)

Let, u=x \(\int {dv = \int {\cos 5xdx} } \)

du=dx \(v = \frac{{\sin 5x}}{5}\)

Step 2- Integrating the equation

\(\int {x\cos 5xdx = \frac{x}{5}} \sin 5x - \int {\frac{{\sin 5x}}{5}} dx\)

=\(\frac{x}{5}\sin 5x + \frac{{\cos 5x}}{{5(5)}} + c\)

=\(\frac{x}{5}\sin 5x + \frac{{\cos 5x}}{{25}} + c\)

Hence , \(\int {x\cos 5x = } \frac{x}{5}\sin 5x + \frac{{\cos 5x}}{{25}} + c\)