Calculus Examples

$\int \sqrt[3]{8 - 9 x^{2}} (- 18 x) d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 - 9 x^{2}}$ as ${(8 - 9 x^{2})}^{\frac{1}{3}}$ .

$\frac{d}{d x} [\int {(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x) d x]$

Step 2

$\int {(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x) d x$ is an antiderivative of ${(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x)$ , so by definition $\frac{d}{d x} [\int {(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x) d x]$ is ${(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x)$ .

${(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x)$

Step 3

Reorder the factors of ${(8 - 9 x^{2})}^{\frac{1}{3}} (- 18 x)$ .

$- 18 x {(8 - 9 x^{2})}^{\frac{1}{3}}$