Calculus Examples

$\int (5 x^{2} - 3) (2 x) d x$

Step 1

Move $2$ to the left of $5 x^{2} - 3$ .

$\int 2 (5 x^{2} - 3) x d x$

Step 2

Let $u = 2 (5 x^{2} - 3)$ . Then $d u = 20 x d x$ , so $\frac{1}{20} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 (5 x^{2} - 3)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 (5 x^{2} - 3)$ .

$\frac{d}{d x} [2 (5 x^{2} - 3)]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 (5 x^{2} - 3)$ with respect to $x$ is $2 \frac{d}{d x} [5 x^{2} - 3]$ .

$2 \frac{d}{d x} [5 x^{2} - 3]$

Step 2.1.3

By the Sum Rule, the derivative of $5 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 3]$ .

$2 (\frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [- 3])$

Step 2.1.4

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$2 (5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3])$

Step 2.1.5

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

$2 (5 (2 x) + \frac{d}{d x} [- 3])$

Step 2.1.6

Multiply $2$ by $5$ .

$2 (10 x + \frac{d}{d x} [- 3])$

Step 2.1.7

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$2 (10 x + 0)$

Step 2.1.8

Simplify the expression.

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Step 2.1.8.1

Add $10 x$ and $0$ .

$2 (10 x)$

Step 2.1.8.2

Multiply $10$ by $2$ .

$20 x$

Step 2.2

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

$\int u \frac{1}{20} d u$

Step 3

Combine $u$ and $\frac{1}{20}$ .

$\int \frac{u}{20} d u$

Step 4

Since $\frac{1}{20}$ is constant with respect to $u$ , move $\frac{1}{20}$ out of the integral.

$\frac{1}{20} \int u d u$

Step 5

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

$\frac{1}{20} (\frac{1}{2} u^{2} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{20} (\frac{1}{2} u^{2} + C)$ as $\frac{1}{20} \cdot \frac{1}{2} u^{2} + C$ .

$\frac{1}{20} \cdot \frac{1}{2} u^{2} + C$

Step 6.2

Rewrite $\frac{1}{20} \cdot \frac{1}{2} u^{2} + C$ as $\frac{1}{40} u^{2} + C$ .

$\frac{1}{40} u^{2} + C$

Step 7

Replace all occurrences of $u$ with $2 (5 x^{2} - 3)$ .

$\frac{1}{40} {(2 (5 x^{2} - 3))}^{2} + C$