Question

Assertion :If $l_n=\int {\cot }^{n}xdx$ then $5(I_6+I_4)=-\cot x$ Reason: If $l_n=\int {\cot }^{n}xdx$ then $l_n=-\frac{{\cot }^{n-1}x}{n}-I_{n-2}$ where $n\ge 2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is D Both Assertion and Reason are incorrect

$I_n=\int {\cot }^{n}xdx=\int {\cot }^{n-2}x{\cot }^{2}xdx=\int {\cot }^{n-2}x({\csc }^{2}x-1)dx=\int {\cot }^{n-2}x{\csc }^{2}xdx-I_{n-2}$
$I_n+I_{n-2}=\frac{{\cot }^{n-1}x}{n-1}$
$\therefore 5(I_6+I_4)={\cot }^{5}x$