Answer to Question #136956 in Calculus for Sagar

Question

Integrate x2+x+5/(x2+4)(x+1) dx

Answer 1

Solution:

$\int \frac{x^2+x+5}{\left(x^2+4\right)\left(x+1\right)}dx$

$=\int \frac{(x^2+4)+(x+1)}{\left(x^2+4\right)\left(x+1\right)}dx$

$=\int (\frac{1}{x^2+4}+\frac{1}{x+1})dx$

$=\int \frac{1}{x^2+4}dx+\int \frac{1}{x+1}dx$

$=\int \frac{1}{x^2+2^2}dx+\int \frac{1}{x+1}dx$

On integrating,

$=\frac{1}{2}\tan^{-1} \left(\frac{x}{2}\right)+\ln \left|x+1\right|+C$ [Using $\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\tan^{-1} \left(\frac{x}{a}\right) ;\int \frac{1}{x}dx=\ln|x|$ ]

Answer:

$\frac{1}{2}\tan^{-1} \left(\frac{x}{2}\right)+\ln \left|x+1\right|+C$