Question

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

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

Answer 1

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

$\int \frac{dx}{x^{1/5}{(1+x^{4/5})}^{1/2}}$
Put $1+x^{4/5}=t$
$\frac{4}{5}{x}^{-1/5}dx=dt$
$\frac{dx}{x^{1/5}}=\frac{5}{4}dt$
So, $I=\frac{5}{4}\int \frac{dt}{t^{1/2}}$
$I=\frac{5}{2}{t}^{1/2}+C$
$I=\frac{5}{2}(1+x^{4/5}{)}^{1/2}+C$