Question

Evaluate: $\int \frac{dx}{x^{1/5}(1+x^{4/5}{)}^{1/2}}$ is

• A. $\sqrt{1+x^{4/5}}+k$
• B. $\frac{5}{2}\sqrt{1+x^{4/5}}+k$
• C. $x^{5/5}(1+x^{4/5}{)}^{1/2}+k$
• D. $x^{1/5}(1+x^{1/5}{)}^{1/2}+k$

Answer 1

The correct option is B $\frac{5}{2}\sqrt{1+x^{4/5}}+k$

$\begin{array}{c}\int \frac{1}{x^{\frac{1}{5}}\sqrt{1+x^{\frac{4}{5}}}}dx\\ Applyu-substitutionu=1+x^{\frac{4}{5}}\\ =\int \frac{5}{4\sqrt{u}}\\ =\frac{5}{4}\int u\frac{-1}{z}du\\ =\frac{5}{4}.\frac{u^{\frac{-1}{z}+1}}{\frac{-1}{2}+1}\\ =\frac{5}{4}.\frac{\left(1+x^{\frac{4}{5}}\right)-\frac{1}{z}+1}{\frac{-1}{2}+1}\\ =\frac{5}{2}{\left(1+x^{\frac{4}{5}}\right)}^{\frac{1}{z}}\\ =\frac{5}{2}{\left(1+x^{\frac{4}{5}}\right)}^{\frac{1}{z}}+c\end{array}$