First slide

Methods of integration

Question

$If \int \frac{d x}{x^{2} {(x^{n} + 1)}^{(n - 1) / n}} = - [f (x)]^{1 / n} + c, then f (x) is$

Moderate

A

$(1 + x^{n})$
B

$1 + x^{- n}$
C

$x^{n} + x^{- n}$
D

$none of these$

Solution

$We have \int \frac{d x}{x^{2} {(x^{n} + 1)}^{(n - 1) / n}} = \int \frac{d x}{x^{2} x^{n - 1} {(1 + \frac{1}{x^{n}})}^{(n - 1) / n}}$
$= \int \frac{d x}{x^{n + 1} {(1 + x^{- n})}^{(n - 1) / n}}$
$Put 1 + x^{- n} = t$
$\begin{matrix} ∴ - n x^{- n - 1} d x = d t or \frac{d x}{x^{n + 1}} = - \frac{d t}{n} \\ ∴ \int \frac{d x}{x^{2} {(x^{n} + 1)}^{(n - 1) / n}} = - \frac{1}{n} \int \frac{d t}{t^{(n - 1) / n}} \\ = - \frac{1}{n} \int {}_{t}{\frac{1 - n}{n}} = - \frac{1}{n} \frac{t^{\frac{1 - n}{n} + 1}}{\frac{1 - n}{n} + 1} + C \\ = - t^{1 / n} + C = - {(1 + x^{- n})}^{1 / n} + C \end{matrix}$

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