Evaluate $\int \sin^{-1} \left( \frac{2x+2}{\sqrt{4x^2+8x+13}} \right) dx$. If the result is express

Question

Evaluate $\int \sin^{-1} \left( \frac{2x+2}{\sqrt{4x^2+8x+13}} ight) dx$. If the result is expressed in the form $P(x) 	an^{-1}\left(\frac{2x+2}{3}ight) - Q \ln(4x^2+8x+13) + C$, where $P(x)$ is a polynomial and $Q$ is a rational number, find the value of $P(1) + 4Q$.

Professor PC Thomas Classes · Answer

Note that $4x^2+8x+13 = 4(x+1)^2+9$. Let $2(x+1) = 3 	an 	heta$, then $2 dx = 3 \sec^2 	heta d	heta$. The expression inside the inverse sine becomes $\frac{3 	an 	heta}{\sqrt{9	an^2	heta+9}} = \sin	heta$. Thus, the integral is $\int 	heta \cdot \frac{3}{2}\sec^2	heta d	heta$. Using by-parts, we get $\frac{3}{2}(	heta	an	heta - \ln|\sec	heta|) + C$. Substituting back, we get $(x+1)	an^{-1}\left(\frac{2x+2}{3}ight) - \frac{3}{4}\ln(4x^2+8x+13) + C$. Here $P(x) = x+1$ and $Q = 3/4$. Thus, $P(1) + 4Q = 2 + 3 = 5$.