Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} + 3$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

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

Step 2

Simplify the expression.

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Step 2.1

Simplify.

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Step 2.1.1

Rewrite ${\sqrt{u - 3}}^{2}$ as $u - 3$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.1.3

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

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

Step 2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.4.1

Cancel the common factor.

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

Step 2.1.1.4.2

Rewrite the expression.

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

Step 2.1.1.5

Simplify.

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

Step 2.1.2

Combine $\frac{1}{2}$ and $\sqrt{u}$ .

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

Raise $u$ to the power of $1$ .

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

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5

Write $1$ as a fraction with a common denominator.

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 3.7

Add $2$ and $1$ .

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

Step 3.8

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

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

Step 4

Move the negative in front of the fraction.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

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

Step 8

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

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

Step 9

Since $\frac{3}{2}$ is constant with respect to $u$ , move $\frac{3}{2}$ out of the integral.

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}} - \frac{3}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 11.2

Rewrite $\frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}} - \frac{3}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ as $\frac{1}{5} u^{\frac{5}{2}} - \frac{3}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$ .

$\frac{1}{5} u^{\frac{5}{2}} - \frac{3}{2} \cdot \frac{2}{3} u^{\frac{3}{2}} + C$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $\frac{2}{3}$ by $\frac{3}{2}$ .

$\frac{1}{5} u^{\frac{5}{2}} - \frac{2 \cdot 3}{3 \cdot 2} u^{\frac{3}{2}} + C$

Step 11.3.2

Multiply $2$ by $3$ .

$\frac{1}{5} u^{\frac{5}{2}} - \frac{6}{3 \cdot 2} u^{\frac{3}{2}} + C$

Step 11.3.3

Multiply $3$ by $2$ .

$\frac{1}{5} u^{\frac{5}{2}} - \frac{6}{6} u^{\frac{3}{2}} + C$

Step 11.3.4

Cancel the common factor of $6$ .

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Step 11.3.4.1

Cancel the common factor.

$\frac{1}{5} u^{\frac{5}{2}} - \frac{6}{6} u^{\frac{3}{2}} + C$

Step 11.3.4.2

Rewrite the expression.

$\frac{1}{5} u^{\frac{5}{2}} - 1 \cdot 1 u^{\frac{3}{2}} + C$

Step 11.3.5

Multiply $- 1$ by $1$ .

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

Step 12

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

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