Exercise
$\int\frac{x+1}{\left(4x^2-1\right)\left(4+9x^2\right)}dx$
Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int((x+1)/((4x^2-1)(4+9x^2)))dx. Rewrite the fraction \frac{x+1}{\left(4x^2-1\right)\left(4+9x^2\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{\frac{4}{25}x+\frac{4}{25}}{4x^2-1}+\frac{-\frac{9}{25}x-\frac{9}{25}}{4+9x^2}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{\frac{4}{25}x+\frac{4}{25}}{4x^2-1}dx results in: \frac{1}{50}\ln\left(x^2-\frac{1}{4}\right)-\frac{1}{25}\ln\left(2x+1\right)+\frac{1}{25}\ln\left(2x-1\right). Gather the results of all integrals.
Find the integral int((x+1)/((4x^2-1)(4+9x^2)))dx
Final answer to the exercise
$-\frac{1}{25}\ln\left|2x+1\right|+\frac{1}{25}\ln\left|2x-1\right|+\frac{1}{50}\ln\left|x^2-\frac{1}{4}\right|-\frac{3}{50}\arctan\left(\frac{3x}{2}\right)-\frac{1}{25}\ln\left|\sqrt{4+9x^2}\right|+C_1$