Question

$\int \frac{x+2}{(x^2+3x+2)\sqrt{x+1}}dx$ is equal to

• A. $\frac{1}{\sqrt{3}}{\tan }^{-1}\left(\frac{x}{\sqrt{3(x+1)}}\right)$
• B. $\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{x}{\sqrt{3(x+1)}}\right)$
• C. $\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{x}{\sqrt{x+1}}\right)$
• D. None of these

Answer 1

Answer: (D)

None of these

$\int \frac{x+2}{(x^2+3x+2)\sqrt{x+1}}dx$

$=\int \frac{x+2}{(x+2)(x+1)\sqrt{x+1}}dx$

$=\int \frac{1}{{(x+1)}^{3/2}}dx$

$=\int {(x+1)}^{-3/2}dx$

$=\frac{{(x+1)}^{-1/2}}{-1/2}+c$

$=\frac{-2}{\sqrt{x+1}}+c$ [D]