Exercise
$\int\frac{4x^{2}+2x+8}{x\left(x^{2}+4\right)^{2}}dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int((4x^2+2x+8)/(x(x^2+4)^2))dx. Rewrite the fraction \frac{4x^2+2x+8}{x\left(x^2+4\right)^2} in 3 simpler fractions using partial fraction decomposition. Simplify the expression. The integral \int\frac{1}{2x}dx results in: \frac{1}{2}\ln\left(x\right). The integral \int\frac{2x+2}{\left(x^2+4\right)^2}dx results in: \frac{1}{-x^2-4}+\frac{1}{8}\arctan\left(\frac{x}{2}\right)+\frac{x}{4\left(x^2+4\right)^{\left(\frac{1}{2}+\frac{1}{2}\right)}}.
Find the integral int((4x^2+2x+8)/(x(x^2+4)^2))dx
Final answer to the exercise
$\frac{1}{2}\ln\left|x\right|+\frac{x}{4\left(x^2+4\right)}+\frac{1}{8}\arctan\left(\frac{x}{2}\right)+\frac{1}{-x^2-4}-\frac{1}{2}\ln\left|\sqrt{x^2+4}\right|+C_1$