$\int \frac{1}{x^{1 / 5}\left(1+x^{4 / 5

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Indefinite Integration

Question:

$\int \frac{1}{x^{1 / 5}\left(1+x^{4 / 5}\right)^{1 / 2}} d x$ , is

Options:

$\sqrt{1+x^{4 / 5}}+C$

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

$x^{4 / 5} \sqrt{1+x^{4 / 5}}+C$

none of these

Correct Answer:

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

Explanation:

Let $I=\int \frac{1}{x^{1 / 5}\left(1+x^{4 / 5}\right)^{1 / 2}} d x$

$\Rightarrow I=\frac{5}{4} \int \frac{1}{\left(1+x^{4 / 5}\right)^{1 / 2}}\left(\frac{4}{5} x^{-1 / 5}\right) d x$

$\Rightarrow I=\frac{5}{4} \int \frac{1}{\left(1+x^{4 / 5}\right)^{1 / 2}} d\left(1+x^{4 / 5}\right)=\frac{5}{2}\left(1+x^{4 / 5}\right)^{1 / 2}+C$