$\int \dfrac{dx}{x^2+2x+2}$ equals
A$x\tan^{-1}(x+1)+C$
B$\tan^{-1}(x+1)+C$
C$(x+1)\tan^{-1}x+C$
D$\tan^{-1}x+C$
Answer & Solution
Correct answer: B. $\tan^{-1}(x+1)+C$
1. Complete the square: $x^2+2x+2=(x+1)^2+1$.
2. Put $t=x+1$, $dt=dx$.
3. Integral $=\int \dfrac{dt}{t^2+1}=\tan^{-1}t$.
4. Back-substitute: $\tan^{-1}(x+1)+C$. Differentiating confirms B.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.314_
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