Mathematics - Indefinite Integration Question with Solution | TestHub

Here, $\begin{array}{rlrl}& \text{ and }& ff(x)& =\frac{f(x)}{{\left[1+f(x{)}^n\right]}^{1/n}}=\frac{x}{{\left(1+2x^n\right)}^{1/n}}\\ & \therefore fff(x)& =\frac{x}{{\left(1+3x^n\right)}^{1/n}}& \\ & \text{ Let }& g(x)& =\underset{n\text{ times }}{\underbrace{(f\circ fo\dots of)}}(x)=\frac{x}{{\left(1+n{x}^n\right)}^{1/n}}\\ I& =\int x^{n-2}g(x)dx=\int \frac{x^{n-1}dx}{{\left(1+n{x}^n\right)}^{1/n}}& & \\ & & =\frac{1}{n^2}\int \frac{n^2{x}^{n-1}dx}{{\left(1+n{x}^n\right)}^{1/n}}=\frac{1}{n^2}\int \frac{\frac{d}{dx}\left(1+n{x}^n\right)}{{\left(1+n{x}^n\right)}^{1/n}}dx& \end{array}$ $\therefore I=\frac{1}{n(n-1)}{\left(1+n{x}^n\right)}^{1-\frac{1}{n}}+k$.