Calculus Examples

$\int {(3 x^{2} - 4 x + 2)}^{4} (3 x - 2) d x$

Step 1

Let $u = 3 x^{2} - 4 x + 2$ . Then $d u = (6 x - 4) d x$ , so $\frac{1}{2} d u = (3 x - 2) d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 3 x^{2} - 4 x + 2$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $3 x^{2} - 4 x + 2$ .

$\frac{d}{d x} [3 x^{2} - 4 x + 2]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x^{2} - 4 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 (2 x) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.1.3.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 4 x]$ .

Tap for more steps...

Step 1.1.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$6 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.1.4.2

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

$6 x - 4 \cdot 1 + \frac{d}{d x} [2]$

Step 1.1.4.3

Multiply $- 4$ by $1$ .

$6 x - 4 + \frac{d}{d x} [2]$

Step 1.1.5

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.5.1

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

$6 x - 4 + 0$

Step 1.1.5.2

Add $6 x - 4$ and $0$ .

$6 x - 4$

Step 1.2

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

$\int u^{4} \frac{1}{2} d u$

Step 2

Combine $u^{4}$ and $\frac{1}{2}$ .

$\int \frac{u^{4}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int u^{4} d u$

Step 4

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{2} (\frac{1}{5} u^{5} + C)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Rewrite $\frac{1}{2} (\frac{1}{5} u^{5} + C)$ as $\frac{1}{2} \cdot \frac{1}{5} u^{5} + C$ .

$\frac{1}{2} \cdot \frac{1}{5} u^{5} + C$

Step 5.2

Rewrite $\frac{1}{2} \cdot \frac{1}{5} u^{5} + C$ as $\frac{1}{10} u^{5} + C$ .

$\frac{1}{10} u^{5} + C$

Step 6

Replace all occurrences of $u$ with $3 x^{2} - 4 x + 2$ .

$\frac{1}{10} {(3 x^{2} - 4 x + 2)}^{5} + C$