Calculus Examples

$\int \frac{12 x + 18}{\sqrt{x^{2} + 3 x - 4}} d x$

Step 1

Simplify.

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Step 1.1

Factor $6$ out of $12 x + 18$ .

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Step 1.1.1

Factor $6$ out of $12 x$ .

$\int \frac{6 (2 x) + 18}{\sqrt{x^{2} + 3 x - 4}} d x$

Step 1.1.2

Factor $6$ out of $18$ .

$\int \frac{6 (2 x) + 6 (3)}{\sqrt{x^{2} + 3 x - 4}} d x$

Step 1.1.3

Factor $6$ out of $6 (2 x) + 6 (3)$ .

$\int \frac{6 (2 x + 3)}{\sqrt{x^{2} + 3 x - 4}} d x$

Step 1.2

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.2.2

Write the factored form using these integers.

$\int \frac{6 (2 x + 3)}{\sqrt{(x - 1) (x + 4)}} d x$

Step 2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{2 x + 3}{\sqrt{(x - 1) (x + 4)}} d x$

Step 3

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

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Step 3.1

Let $u = (x - 1) (x + 4)$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $(x - 1) (x + 4)$ .

$\frac{d}{d x} [(x - 1) (x + 4)]$

Step 3.1.2

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 $f (x) = x - 1$ and $g (x) = x + 4$ .

$(x - 1) \frac{d}{d x} [x + 4] + (x + 4) \frac{d}{d x} [x - 1]$

Step 3.1.3

Differentiate.

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Step 3.1.3.1

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

$(x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [4]) + (x + 4) \frac{d}{d x} [x - 1]$

Step 3.1.3.2

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

$(x - 1) (1 + \frac{d}{d x} [4]) + (x + 4) \frac{d}{d x} [x - 1]$

Step 3.1.3.3

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$(x - 1) (1 + 0) + (x + 4) \frac{d}{d x} [x - 1]$

Step 3.1.3.4

Simplify the expression.

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Step 3.1.3.4.1

Add $1$ and $0$ .

$(x - 1) \cdot 1 + (x + 4) \frac{d}{d x} [x - 1]$

Step 3.1.3.4.2

Multiply $x - 1$ by $1$ .

$x - 1 + (x + 4) \frac{d}{d x} [x - 1]$

Step 3.1.3.5

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$x - 1 + (x + 4) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 3.1.3.6

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

$x - 1 + (x + 4) (1 + \frac{d}{d x} [- 1])$

Step 3.1.3.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$x - 1 + (x + 4) (1 + 0)$

Step 3.1.3.8

Simplify by adding terms.

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Step 3.1.3.8.1

Add $1$ and $0$ .

$x - 1 + (x + 4) \cdot 1$

Step 3.1.3.8.2

Multiply $x + 4$ by $1$ .

$x - 1 + x + 4$

Step 3.1.3.8.3

Add $x$ and $x$ .

$2 x - 1 + 4$

Step 3.1.3.8.4

Add $- 1$ and $4$ .

$2 x + 3$

Step 3.2

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

$6 \int u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$6 \int u^{- \frac{1}{2}} d u$

Step 5

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

$6 (2 u^{\frac{1}{2}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $6 (2 u^{\frac{1}{2}} + C)$ as $6 \cdot 2 u^{\frac{1}{2}} + C$ .

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

Step 6.2

Multiply $6$ by $2$ .

$12 u^{\frac{1}{2}} + C$

Step 7

Replace all occurrences of $u$ with $(x - 1) (x + 4)$ .

$12 {((x - 1) (x + 4))}^{\frac{1}{2}} + C$