Question

Evaluate :${\int }_{-10}^{10}\log \left[\frac{\left(a+x\right)}{\left(a-x\right)}\right]dx$

• A. $0$
• B. $-2\log \left(a+10\right)$
• C. $2\log \left[\left(a+x\right)/\left(a-x\right)\right]$
• D. $2\log \left(a+x\right)$

Answer 1

The correct option is A

$0$

Evaluate the given integral :

Consider the given equation as

$I={\int }_{-10}^{10}\log \left(\frac{a+x}{a-x}\right)dx$

In the given Equation both the limits are same but only differ in their sign so that we have to check whether the given function is even or odd.

We know that, $f\left(-x\right)=f\left(x\right)$ for even function and $f\left(-x\right)=-f\left(x\right)$ for odd function.

$$\begin{gathered} f\left(x\right)=\log \left(\frac{a-x}{a+x}\right) \\ f\left(-x\right)=\log \left(\frac{a-x}{a+x}\right) \\ f\left(-x\right)=-\log \left(\frac{a+x}{a-x}\right) \\ f\left(-x\right)=-f\left(x\right) \end{gathered}$$

Hence, the function $f\left(x\right)$ is odd function

According to the property of definite integral ${\int }_{-a}^{a}f\left(x\right)dx=0$ if $f\left(x\right)$ is an odd function.

Hence, $I={\int }_{-10}^{10}\log \left(\frac{a+x}{a-x}\right)dx=0$

Hence, the correct answer is Option (A).