Mathematics - Indefinite Integration Question with Solution | TestHub

We have,$f\left(x\right)=\int \frac{5x^8+7x^6}{{\left(x^2+1+2x^7\right)}^2} dx$

$=\int \frac{5x^8+7x^6}{x^{14}{\left(\frac{1}{x^5}+\frac{1}{x^7}+2\right)}^2}dx$

$=\int \frac{\frac{5}{x^6}+\frac{7}{x^8}}{{\left(\frac{1}{x^5}+\frac{1}{x^7}+2\right)}^2}dx$

Put,$\frac{1}{x^5}+\frac{1}{x^7}+2=t$

$\Rightarrow \left(-\frac{5}{x^6}-\frac{7}{x^8}\right)dx=dt$

$\therefore f\left(x\right)=-\int \frac{dt}{t^2}=\frac{1}{t}+c$, where$c$is the constant of integration.

$=\frac{x^7}{x^2+1+2x^7}+c$

Now,$f\left(0\right)=0$$\Rightarrow c=0$

$\therefore f\left(1\right)=\frac{1}{1^2+2+{2.1}^7}=\frac{1}{4}$