Evaluate: (x^4-x )^ 1 4 x^5 d x - Vidyadip

MCQ

Evaluate: $\int \frac{\left(x^4-x\right)^{\frac{1}{4}}}{x^5} d x$

✓
$\frac{4}{15}\left(1-\frac{1}{x^3}\right)^{\frac{5}{4}}+C$
B
$\frac{-4}{15}\left(1-\frac{1}{x^3}\right)^{\frac{5}{4}}+C$
C
$\frac{2}{15}\left(1-\frac{1}{x^3}\right)^{\frac{5}{4}}+C$
D
$\frac{-2}{15}\left(1-\frac{1}{x^3}\right)^{\frac{5}{4}}+C$

✓

Answer

Correct option: A.

$\frac{4}{15}\left(1-\frac{1}{x^3}\right)^{\frac{5}{4}}+C$

(a) : Let $I=\int \frac{\left(x^4-x\right)^{\frac{1}{4}}}{x^5} d x$
$
\Rightarrow \quad I=\int \frac{x\left(1-\frac{1}{x^3}\right)^{\frac{1}{4}}}{x^5} d x=\int \frac{\left(1-\frac{1}{x^3}\right)^{\frac{1}{4}}}{x^4} d x
$
Put $1-\frac{1}{x^3}=t \Rightarrow \frac{3}{x^4} d x=d t$

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