Integration Strategy — Question 7

Question 7

Choose the best method and evaluate: $$\int\frac{x^2}{x^3+8}\,dx.$$

Question 7 – Solution

Step 1: Identify the structure. The denominator is $x^3+8$, and its derivative is $3x^2$. The numerator contains $x^2$, so substitution is the shortest method.

Step 2: Substitute $u=x^3+8$. Then $$du=3x^2\,dx,\qquad x^2\,dx=\frac13du.$$ Step 3: Rewrite and integrate. $$\begin{align*} \int\frac{x^2}{x^3+8}\,dx &=\frac13\int\frac{du}{u}\\ &=\frac13\ln|u|+C. \end{align*}$$ Step 4: Return to $x$. $$\boxed{\frac13\ln|x^3+8|+C}$$