Calculus Examples

$\int 3 x^{2} - \sqrt{5 x} + 2 d x$

Step 1

Split the single integral into multiple integrals.

$\int 3 x^{2} d x + \int - \sqrt{5 x} d x + \int 2 d x$

Step 2

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

$3 \int x^{2} d x + \int - \sqrt{5 x} d x + \int 2 d x$

Step 3

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

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

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 5

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 5.1.2

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

$5 \frac{d}{d x} [x]$

Step 5.1.3

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

$5 \cdot 1$

Step 5.1.4

Multiply $5$ by $1$ .

$5$

Step 5.2

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

$3 (\frac{1}{3} x^{3} + C) - \int \sqrt{u} \frac{1}{5} d u + \int 2 d x$

Step 6

Simplify.

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Step 6.1

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

$3 (\frac{x^{3}}{3} + C) - \int \sqrt{u} \frac{1}{5} d u + \int 2 d x$

Step 6.2

Combine $\sqrt{u}$ and $\frac{1}{5}$ .

$3 (\frac{x^{3}}{3} + C) - \int \frac{\sqrt{u}}{5} d u + \int 2 d x$

Step 7

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$3 (\frac{x^{3}}{3} + C) - (\frac{1}{5} \int \sqrt{u} d u) + \int 2 d x$

Step 8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify.

$x^{3} - \frac{2 u^{\frac{3}{2}}}{15} + 2 x + C$

Step 12

Replace all occurrences of $u$ with $5 x$ .

$x^{3} - \frac{2 {(5 x)}^{\frac{3}{2}}}{15} + 2 x + C$

Step 13

Reorder terms.

$x^{3} - \frac{2}{15} {(5 x)}^{\frac{3}{2}} + 2 x + C$