Q. 75 - Calculus

Question

Solve each of the integrals in Exercises $75–78$ by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

$\int \frac{x^3}{x^2+4}dx$

Step-by-Step Solution

Answer

The value of integral is $\frac{x^2}{2}-2\ln \left(\left|x^2+4\right|\right)+C$

Step 1. Given information

An expression is given as $\int \frac{x^3}{x^2+4}dx$

Step 2. Solving integral

First split the term using long distance method and then simplify as,

$$\begin{gathered} \int \left(x-\frac{4x}{x^2+4}\right)dx \\ =\int xdx-\int \frac{4x}{x^2+4}dx \\ =\frac{x^2}{2}-\int \frac{4x}{x^2+4}dx \\ =\frac{x^2}{2}-\int \frac{2}{u}dx \\ =\frac{x^2}{2}-2\ln \left(\right|u\left|\right)+C \\ =\frac{x^2}{2}-2\ln \left(\left|x^2+4\right|\right)+C \end{gathered}$$