$$\int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)}$$ equals ______

Category: Questions$$\int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)}$$ equals ______
Editor">Editor Staff asked 11 months ago

$$\int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)}$$ equals ______

(a) $$\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{5} tan^{-1}x+5C$$

(b) $$\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{6} tan^{-1}x+C$$

(c) $$\frac{3}{5}log|x+2| + \frac{1}{6}log|x^2+1|+\frac{1}{6} tan^{-1}x+C$$

(d) $$\frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{5} tan^{-1}x+C$$

This question was addressed to me during an online interview.

My doubt stems from Integration by Partial Fractions topic in division Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options
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Editor">Editor Staff answered 11 months ago

Correct option is (d) \frac{3}{5}log|x+2| + \frac{1}{5}log|x^2+1|+\frac{1}{5} tan^{-1}x+C

Explanation: \int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)} = \frac{A}{(x+2)} + \frac{Bx+C}{(x^2+1)}

Now equating, (x^2+x+1) = A (x^2+1) + (Bx+C) (x+2)

After equating and solving for coefficient we get values,

A=\frac{3}{5}, B=\frac{2}{5}, C=\frac{1}{5}, now putting these values in the equation we get,

\int \frac{(x^2+x+1)dx}{(x+2)(x^2+1)} = \frac{3}{5} \int \frac{dx}{(x+2)} + \frac{1}{5} \int \frac{2xdx}{(x^2+1)} + \frac{1}{5} \int \frac{dx}{(x^2+1)}

Hence it comes, \frac{3}{5} log|x+2| + \frac{1}{5} log|x^2+1|+\frac{1}{5}tan^{-1}x+C