Question

${\int }_{-5}^5\log \left(\frac{130-x^3}{130+x^3}\right)dx$ is equal to

• A. $log\frac{57}{5}$
• B. $2{\int }_{-5}^5\log \left(\frac{130-x^3}{130+x^3}\right)dx$
• C. $0$
• D. $log\frac{5}{57}$

Answer 1

Answer: (C)

$0$

Let $f\left(x\right)=log\left(\frac{130-x^3}{130+x^3}\right)$

$f(-x)=log\left(\frac{130+x^3}{130-x^3}\right)$

$f(-x)=-log\left(\frac{130-x^3}{130+x^3}\right)=-f\left(x\right)$

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

${\int }_{-a}^{a}f\left(x\right)dx=0$ if f(x) is an odd function

${\int }_{-5}^{5}log\left(\frac{130-x^3}{130+x^3}\right)dx=0$