Question

Assertion :$\int \frac{x+3}{\sqrt{5-4x-x^2}}=-\sqrt{5-4x-x^2}+{\sin }^{-1}\left(\frac{x+2}{3}\right)+C$ Reason: $\int \frac{dx}{\sqrt{a^2-x^2}}=\frac{x^2\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}{\sin }^{-1}\left(\frac{x}{a}\right)$

• 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 C Assertion is correct but Reason is incorrect

$\int \frac{x+3}{\sqrt{5-4x-x^2}}dx$

$=\int -\frac{1}{2}\frac{(-2x-6)}{\sqrt{5-4x-x^2}}dx=\int -\frac{1}{2}\frac{(-2x-4-2)}{\sqrt{5-4x-x^2}}dx$

$=\int -\frac{1}{2}\frac{(-2x-4)}{\sqrt{5-4x-x^2}}dx+\int \frac{dx}{\sqrt{5-4x-x^2}}$

$=-\sqrt{5-4x-x^2}+\int \frac{dx}{\sqrt{9-{(x+2)}^2}}$
$=-\sqrt{5-4x-x^2}+{\sin }^{-1}\left(\frac{x+2}{3}\right)+C$
$\therefore$ Assertion (A) is true
And
$\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\left(\frac{x}{a}\right)+C$
$\therefore$ Reason (R) is false.