Mathematics - Definite Integration Question with Solution | TestHub

$$\begin{array}{rl}& {\int }_{-0.9}^{0.9}\left\{\left[x^2\right]+\log \left(\frac{2-x}{2+x}\right)\right\}dx\\ =& {\int }_{-0.9}^{0.9}\left[x^2\right]dx+{\int }_{-0.9}^{0.9}\log \left(\frac{2-x}{2+x}\right)dx\end{array}$$ $$\begin{array}{rl}& =0+{\int }_{-0.9}^{0.9}\log \left(\frac{2-x}{2+x}\right)dx\\ & \text{ Put }x=-x\Rightarrow f(x)=\log \frac{2-x}{2+x}\\ & \text{ and }f(-x)=\log \frac{2+x}{2-x}\\ & =-\log \frac{(2-x)}{2+x}=-f(x)\end{array}$$ So, it is an odd function, hence Required integral $=0$.