Question

Evaluate the integral\(\int_{{\rm{ - 3}}}^{\rm{3}} {\frac{{\rm{x}}}{{{\rm{1 + |x|}}}}} {\rm{dx}}\).

Short answer

The integral value of the given equation is\(\int_{{\rm{ - 3}}}^{\rm{3}} {\frac{{\rm{x}}}{{{\rm{1 + |x|}}}}} {\rm{dx = 0}}\).

Step-by-step solution

Expand the equation.

Evaluate the equation.

\(\begin{aligned}{c}\int_{{\rm{ - 3}}}^{\rm{3}} {\frac{{\rm{x}}}{{{\rm{1 + |x|}}}}} {\rm{dx = ( - x - ln|1 - x|)}}_{{\rm{ - 3}}}^{\rm{0}}{\rm{ + (x - ln|1 + x|)}}_{\rm{0}}^{\rm{3}}\\{\rm{ = - ( - ( - 3) - ln|1 - ( - 3)|) + (3 - ln|1 + 3|)}}\\{\rm{ = - 3 + ln4 + 3 - ln4}}\\{\rm{ = 0}}\end{aligned}\)

Therefore, the integral value of the given equation is\(\int_{{\rm{ - 3}}}^{\rm{3}} {\frac{{\rm{x}}}{{{\rm{1 + |x|}}}}} {\rm{dx = 0}}\).