Question

Evaluate the indefinite integral.

\(\int x \sin \left( {{x^2}} \right)dx\)

Short answer

The value of \(\int x \sin \left( {{x^2}} \right)dx\)is \(I = - \frac{1}{2}\left( {\cos {x^2}} \right) + c\).

Step-by-step solution

Step 1:Definition of the indefinite integral

An integral that has no upper bound and no lower bound is known as indefinite integral.

Evaluating the integral.

The given information is expressed as,

\(I = \int x \sin \left( {{x^2}} \right)dx\)

Let

\(\begin{aligned}{c}u &= {x^2}\\du &= (2x)dx\\dx &= \frac{{du}}{{2x}}\end{aligned}\)

Step 3: Substitutingthe integral.

Substituting the value of\({\bf{x}}\)and\({\bf{dx}}\)for\({\bf{u}}\)and\({\bf{du}}\)

\(I = \frac{1}{2}\int {(\sin u)} du\)

After Integrating we get

\(\;\;\;I = \frac{1}{2}( - \cos u) + c\)

Substitute again\(u = {x^2}\)

\(I = - \frac{1}{2}\left( {\cos {x^2}} \right) + c\)

Hence the value of \(\int x \sin \left( {{x^2}} \right)dx\)is \(I = - \frac{1}{2}\left( {\cos {x^2}} \right) + c\).