Calculus Examples

$\int \frac{3 x^{2}}{{(3 x^{3} - 2)}^{\frac{2}{3}}} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{x^{2}}{{(3 x^{3} - 2)}^{\frac{2}{3}}} d x$

Step 2

Let $u = 3 x^{3} - 2$ . Then $d u = 9 x^{2} d x$ , so $\frac{1}{9} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.1

Let $u = 3 x^{3} - 2$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 2.1.1

Differentiate $3 x^{3} - 2$ .

$\frac{d}{d x} [3 x^{3} - 2]$

Step 2.1.2

By the Sum Rule, the derivative of $3 x^{3} - 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [- 2]$

Step 2.1.3

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

Tap for more steps...

Step 2.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 (3 x^{2}) + \frac{d}{d x} [- 2]$

Step 2.1.3.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [- 2]$

Step 2.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 2.1.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$9 x^{2} + 0$

Step 2.1.4.2

Add $9 x^{2}$ and $0$ .

$9 x^{2}$

Step 2.2

Rewrite the problem using $u$ and $d u$ .

$3 \int \frac{1}{u^{\frac{2}{3}}} \cdot \frac{1}{9} d u$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Multiply $\frac{1}{u^{\frac{2}{3}}}$ by $\frac{1}{9}$ .

$3 \int \frac{1}{u^{\frac{2}{3}} \cdot 9} d u$

Step 3.2

Move $9$ to the left of $u^{\frac{2}{3}}$ .

$3 \int \frac{1}{9 u^{\frac{2}{3}}} d u$

Step 4

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$3 (\frac{1}{9} \int \frac{1}{u^{\frac{2}{3}}} d u)$

Step 5

Simplify the expression.

Tap for more steps...

Step 5.1

Simplify.

Tap for more steps...

Step 5.1.1

Combine $\frac{1}{9}$ and $3$ .

$\frac{3}{9} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 5.1.2

Cancel the common factor of $3$ and $9$ .

Tap for more steps...

Step 5.1.2.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{9} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 5.1.2.2

Cancel the common factors.

Tap for more steps...

Step 5.1.2.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 1}{3 \cdot 3} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 5.1.2.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 3} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 5.1.2.2.3

Rewrite the expression.

$\frac{1}{3} \int \frac{1}{u^{\frac{2}{3}}} d u$

Step 5.2

Apply basic rules of exponents.

Tap for more steps...

Step 5.2.1

Move $u^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{3} \int {(u^{\frac{2}{3}})}^{- 1} d u$

Step 5.2.2

Multiply the exponents in ${(u^{\frac{2}{3}})}^{- 1}$ .

Tap for more steps...

Step 5.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3} \int u^{\frac{2}{3} \cdot - 1} d u$

Step 5.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

Tap for more steps...

Step 5.2.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$\frac{1}{3} \int u^{\frac{2 \cdot - 1}{3}} d u$

Step 5.2.2.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} \int u^{\frac{- 2}{3}} d u$

Step 5.2.2.3

Move the negative in front of the fraction.

$\frac{1}{3} \int u^{- \frac{2}{3}} d u$

Step 6

By the Power Rule, the integral of $u^{- \frac{2}{3}}$ with respect to $u$ is $3 u^{\frac{1}{3}}$ .

$\frac{1}{3} (3 u^{\frac{1}{3}} + C)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Rewrite $\frac{1}{3} (3 u^{\frac{1}{3}} + C)$ as $\frac{1}{3} \cdot 3 u^{\frac{1}{3}} + C$ .

$\frac{1}{3} \cdot 3 u^{\frac{1}{3}} + C$

Step 7.2

Simplify.

Tap for more steps...

Step 7.2.1

Combine $\frac{1}{3}$ and $3$ .

$\frac{3}{3} u^{\frac{1}{3}} + C$

Step 7.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.2.2.1

Cancel the common factor.

$\frac{3}{3} u^{\frac{1}{3}} + C$

Step 7.2.2.2

Rewrite the expression.

$1 u^{\frac{1}{3}} + C$

Step 7.2.3

Multiply $u^{\frac{1}{3}}$ by $1$ .

$u^{\frac{1}{3}} + C$

Step 8

Replace all occurrences of $u$ with $3 x^{3} - 2$ .

${(3 x^{3} - 2)}^{\frac{1}{3}} + C$