Question

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

• A. $\sqrt{1+x^{4/5}}+c$
• B. $x^{4/5}{(1+x^{4/5})}^{1/2}+c$
• C. $\frac{5}{2}\sqrt{1+x^{4/5}}+c$
• D. None of these

Answer 1

Answer: (C)

$\frac{5}{2}\sqrt{1+x^{4/5}}+c$

Substitute $1+x^{4/5}=t$ so that $\frac{4}{5}{x}^{-1/5}dx=dt$

i.e., $\frac{1}{x^{1/5}}dx=\frac{5}{4}dt$

$\therefore I=\int \frac{\frac{5}{4}dt}{t^{1/2}}=\frac{5}{4}\int t^{-1/2}dt$

$=\frac{5}{4}\frac{\sqrt{t}}{\frac{1}{2}}=\frac{5}{2}\sqrt{1+x^{4/5}}+c$