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