Question

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

• A. $\frac{-2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
• B. $\frac{2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
• C. $\frac{-\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$
• D. $\frac{\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$

Answer 1

The correct option is B $\frac{2\sqrt{x}}{\sqrt[4]{4\sqrt{1+x^2}}}+C$

$\int \frac{dx}{x^{\frac{1}{2}}(1+x^2{)}^{5/4}}$
$\int \frac{dx}{x^3{(1+\frac{1}{x^2})}^{5/4}}$
$1+\frac{1}{x^2}=t$
$\frac{-2}{x^3}dx=dt$
$\frac{dx}{x^3}=\frac{-dt}{2}$
$=\frac{-1}{2}\int \frac{dt}{t^{5/4}}$
$=\frac{2}{t^{1/4}}+c$
$=\frac{2\sqrt{x}}{\sqrt[4]{41+x^2}}+c$