Calculus Examples

$\int \frac{2 x + 2}{\sqrt{3 x^{2} + 6 x}} d x$

Step 1

Simplify.

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Step 1.1

Factor $2$ out of $2 x + 2$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\int \frac{2 (x) + 2}{\sqrt{3 x^{2} + 6 x}} d x$

Step 1.1.2

Factor $2$ out of $2$ .

$\int \frac{2 (x) + 2 (1)}{\sqrt{3 x^{2} + 6 x}} d x$

Step 1.1.3

Factor $2$ out of $2 (x) + 2 (1)$ .

$\int \frac{2 (x + 1)}{\sqrt{3 x^{2} + 6 x}} d x$

Step 1.2

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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Step 1.2.1

Factor $3 x$ out of $3 x^{2}$ .

$\int \frac{2 (x + 1)}{\sqrt{3 x (x) + 6 x}} d x$

Step 1.2.2

Factor $3 x$ out of $6 x$ .

$\int \frac{2 (x + 1)}{\sqrt{3 x (x) + 3 x (2)}} d x$

Step 1.2.3

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

$\int \frac{2 (x + 1)}{\sqrt{3 x (x + 2)}} d x$

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \frac{x + 1}{\sqrt{3 x (x + 2)}} d x$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $3 x (x + 2)$ .

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

Step 3.1.2

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

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

Step 3.1.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x + 2$ .

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

Step 3.1.4

Differentiate.

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Step 3.1.4.1

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

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

Step 3.1.4.2

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

$3 (x (1 + \frac{d}{d x} [2]) + (x + 2) \frac{d}{d x} [x])$

Step 3.1.4.3

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

$3 (x (1 + 0) + (x + 2) \frac{d}{d x} [x])$

Step 3.1.4.4

Simplify the expression.

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Step 3.1.4.4.1

Add $1$ and $0$ .

$3 (x \cdot 1 + (x + 2) \frac{d}{d x} [x])$

Step 3.1.4.4.2

Multiply $x$ by $1$ .

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

Step 3.1.4.5

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

$3 (x + (x + 2) \cdot 1)$

Step 3.1.4.6

Simplify by adding terms.

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Step 3.1.4.6.1

Multiply $x + 2$ by $1$ .

$3 (x + x + 2)$

Step 3.1.4.6.2

Add $x$ and $x$ .

$3 (2 x + 2)$

Step 3.1.5

Simplify.

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Step 3.1.5.1

Apply the distributive property.

$3 (2 x) + 3 \cdot 2$

Step 3.1.5.2

Combine terms.

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Step 3.1.5.2.1

Multiply $2$ by $3$ .

$6 x + 3 \cdot 2$

Step 3.1.5.2.2

Multiply $3$ by $2$ .

$6 x + 6$

Step 3.2

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

$2 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{6} d u$

Step 4

Simplify.

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Step 4.1

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{6}$ .

$2 \int \frac{1}{\sqrt{u} \cdot 6} d u$

Step 4.2

Move $6$ to the left of $\sqrt{u}$ .

$2 \int \frac{1}{6 \sqrt{u}} d u$

Step 5

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$2 (\frac{1}{6} \int \frac{1}{\sqrt{u}} d u)$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $\frac{1}{6}$ and $2$ .

$\frac{2}{6} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.2

Cancel the common factor of $2$ and $6$ .

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Step 6.1.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6} \int \frac{1}{\sqrt{u}} d u$

Step 6.1.2.2

Cancel the common factors.

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Step 6.1.2.2.1

Factor $2$ out of $6$ .

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

Step 6.1.2.2.2

Cancel the common factor.

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

Step 6.1.2.2.3

Rewrite the expression.

$\frac{1}{3} \int \frac{1}{\sqrt{u}} d u$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Replace all occurrences of $u$ with $3 x (x + 2)$ .

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