Question

\[\int\limits_{- \pi/2}^{\pi/2} x \cos^2 x\ dx .\]

 

Answer 1

\[\text{We have}, \]

\[I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} x \cos^2 x\ d x\]

\[Let f\left( x \right) = x \cos^2 x\]

\[ \Rightarrow f\left( - x \right) = \left( - x \right) \cos^2 \left( - x \right)\]

\[ = - x \cos^2 x\]

\[ \therefore f\left( - x \right) = - f\left( x \right)\]

\[i . e . , f\left( x \right) \text{is odd function}\]

\[\text{We know that} \int_{- a}^a f\left( x \right) d x = 0 , \text{if }f\left( x \right) \text{is odd function} . \]

\[ \therefore I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} x \cos^2 x\ d x = 0\]