Mathematics - Definite Integration Question with Solution | TestHub

$I_0=\int dx=x+C$

$$\begin{array}{rl}{I}_1& =\int \cot xdx=\ln |\sin x|+C\\ I_0+I_2& =-\cot x\dots \dots (i)\\ I_3& =\int {\cot }^{3}xdx=\int ({\mathrm{cosec}}^{2}x-1)\cot xdx\\ I_3+I_1& =-\frac{(\cot x{)}^2}{2}\dots \dots (ii)\end{array}$$

Similarly,

$$\begin{array}{rl}{I}_n+I_{n-2}& =\frac{(\cot x{)}^{n-1}}{n-1}\\ I_0+I_1+2(I_2+\dots \dots +I_9)+I_{10}& =-\left(u+\frac{u^2}{2}+\dots +\frac{u^9}{9}\right)\end{array}$$