Question

Evaluate \[ \int \frac{dx}{(x+2)\sqrt{x+1}} \]

• A. \(\tan^{-1}(\sqrt{x+1}) + c\)
• B. \(2\tan^{-1}(\sqrt{x+1}) + c\)
• C. \(2\tan^{-1}(\sqrt{x+2}) + c\)
• D. \(\tan^{-1}(\sqrt{x+2}) + c\)

Hint:

Radical integrals often simplify quickly after an appropriate substitution.

Answer 1

The Correct Option is B

Step 1: Use substitution.
Let \(x + 1 = t^2\). Then \[ dx = 2t\,dt, \quad x + 2 = t^2 + 1 \]
Step 2: Substitute into the integral.
\[ \int \frac{dx}{(x+2)\sqrt{x+1}} = \int \frac{2t\,dt}{(t^2+1)t} = \int \frac{2}{t^2+1}\,dt \]
Step 3: Integrate.
\[ = 2\tan^{-1} t + c \]
Step 4: Substitute back.
\[ = 2\tan^{-1}(\sqrt{x+1}) + c \]