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Solve the following.
Evaluate: ∫ (cos x)/(sin² x + 4sin x + 5) dx
Evaluate: $\int \frac{\cos x}{\sin^2 x + 4\sin x + 5}\,dx$
Answer
Answer
Put $\sin x=t$, so $\cos x\,dx=dt$.
$\int\frac{dt}{t^2+4t+5}=\int\frac{dt}{(t+2)^2+1}$ (completing the square)
$=\tan^{-1}(t+2)+C$
Hence the integral is $\tan^{-1}(\sin x+2)+C$.
Final answer: $\tan^{-1}(\sin x+2)+C$