Exercise
$\int\frac{x^2+9x+2}{\left(x^2+3\right)\left(x^2+5\right)}dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int((x^2+9x+2)/((x^2+3)(x^2+5)))dx. Rewrite the fraction \frac{x^2+9x+2}{\left(x^2+3\right)\left(x^2+5\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{\frac{9}{2}x-\frac{1}{2}}{x^2+3}+\frac{-\frac{9}{2}x+\frac{3}{2}}{x^2+5}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{\frac{9}{2}x-\frac{1}{2}}{x^2+3}dx results in: -\frac{9}{2}\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+\frac{-\arctan\left(\frac{x}{\sqrt{3}}\right)}{2\sqrt{3}}. Gather the results of all integrals.
Find the integral int((x^2+9x+2)/((x^2+3)(x^2+5)))dx
Final answer to the exercise
$\frac{-\arctan\left(\frac{x}{\sqrt{3}}\right)}{2\sqrt{3}}+\frac{9}{2}\ln\left|\sqrt{x^2+3}\right|+\frac{3\arctan\left(\frac{x}{\sqrt{5}}\right)}{2\sqrt{5}}+\frac{9}{2}\ln\left|\frac{\sqrt{5}}{\sqrt{x^2+5}}\right|+C_1$