Question

Assertion :Statement-1: $\int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx={\sec }^{-1}\left|\frac{x^2+1}{x\sqrt{2}}\right|+C$ Reason: staement-2: $\int \frac{dt}{t\sqrt{t^2-a}}=\frac{1}{\sqrt{a}}{\sec }^{-1}\left|\frac{t}{\sqrt{a}}\right|+C$

• 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

Let $I=\int \frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}dx=\int \frac{1-1/x^2}{(x+1/x)\sqrt{x^2+1/x^2}}dx$

Put $t=x+1\Rightarrow dt=dx$

$\Rightarrow I=\int \frac{dt}{t\sqrt{t^2-2}}=\frac{1}{\sqrt{2}}{\sec }^{-1}\frac{t}{\sqrt{2}}+c$ (Using reason)

$=\frac{1}{\sqrt{2}}{\sec }^{-1}\frac{x^2+1}{x\sqrt{2}}+c$