First slide

Methods of integration

Question

Let $f (x) = \frac{x}{{(1 + x^{n})}^{1 / n}}$ for $n \geq 2$ and $g (x) =$
$\frac{f \circ f \dots o f}{n times} .$ Then $\int x^{n - 2} g (x) d x$ equal

Moderate

A

$\frac{1}{n (n - 1)} {(1 + n x^{n})}^{1 - 1 / n} + k$
B

$\frac{1}{n - 1} {(1 + n x^{n})}^{1 - 1 / n} + k$
C

$\frac{1}{n (n - 1)} {(1 + n x^{n})}^{1 + 1 / n} + k$
D

$\frac{1}{n - 1} {(1 + n x^{n})}^{1 + 1 / n} + k$

Solution

We have
$\begin{array}{l} (f \circ f) (x) = f (f (x)) = \frac{f (x)}{{[1 + (f (x))^{n}]}^{1 / n}} \\ = \frac{x}{{(1 + 2 x^{n})}^{1 / n}} \end{array}$
Similarly, $(fofof) (x) = \frac{x}{{(1 + 3 x^{n})}^{1 / n}}$
Continuing in this way we get
$g (x) = \underset{n times}{\underset{⏟}{(fof \dots of) (x)}} = \frac{x}{{(1 + n x^{n})}^{1 / n}}$
Thus , $I = \int x^{n - 2} \frac{x}{{(1 + n x^{n})}^{1 / n}} d x$
Put $1 + n x^{n} = t^{n}$ to obtain
$\begin{array}{l} I = \frac{1}{n} \int \frac{t^{n - 1}}{t} d t = \frac{1}{n} \int t^{n - 2} d t \\ = \frac{1}{n (n - 1)} t^{n - 1} + C = \frac{1}{n (n - 1)} {(1 + n x^{n})}^{1 - 1 / n} + C \end{array}$

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