Question

Evaluate the following definite integral: $\underset{0}{\overset{1}{\int }}\frac{dx}{x+x^{-4}}$

• A. $\log 2$
• B. $\frac{1}{5}\log 3$
• C. $\frac{1}{5}\log 2$
• D. $\frac{1}{2}\log 5$

Answer 1

The correct option is D $\frac{1}{2}\log 5$

Consider, $I={\int }_0^1\frac{1}{x+x^{-4}}dx$

$I={\int }_0^1\frac{x^4}{1+x^5}dx$

$1+x^5=t$

$5x^{4}dx=dt$

$I=\frac{1}{5}{\int }_1^2\frac{1}{t}dt$

$I=\frac{1}{5}{\left[\log 2\right]}_1^2$

$I=\frac{1}{5}\log 2$

Correct option is C