Calculus Examples

$\int \frac{10 x^{3} - 5 x}{\sqrt{x^{4} - x^{2} + 6}} d x$

Step 1

Factor $5 x$ out of $10 x^{3} - 5 x$ .

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Step 1.1

Factor $5 x$ out of $10 x^{3}$ .

$\int \frac{5 x (2 x^{2}) - 5 x}{\sqrt{x^{4} - x^{2} + 6}} d x$

Step 1.2

Factor $5 x$ out of $- 5 x$ .

$\int \frac{5 x (2 x^{2}) + 5 x (- 1)}{\sqrt{x^{4} - x^{2} + 6}} d x$

Step 1.3

Factor $5 x$ out of $5 x (2 x^{2}) + 5 x (- 1)$ .

$\int \frac{5 x (2 x^{2} - 1)}{\sqrt{x^{4} - x^{2} + 6}} d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \frac{x (2 x^{2} - 1)}{\sqrt{x^{4} - x^{2} + 6}} d x$

Step 3

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

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Step 3.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $x^{2}$ .

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

Step 3.1.2

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

$2 x$

Step 3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$5 \int \frac{2 u_{1} - 1}{\sqrt{{\sqrt{u_{1}}}^{4} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4

Simplify.

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Step 4.1

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Step 4.1.1

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

$5 \int \frac{2 u_{1} - 1}{\sqrt{{({u_{1}}^{\frac{1}{2}})}^{4} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.2

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

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{\frac{1}{2} \cdot 4} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.3

Combine $\frac{1}{2}$ and $4$ .

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{\frac{4}{2}} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.4

Cancel the common factor of $4$ and $2$ .

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Step 4.1.4.1

Factor $2$ out of $4$ .

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{\frac{2 \cdot 2}{2}} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $2$ out of $2$ .

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{\frac{2 \cdot 2}{2 (1)}} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.4.2.2

Cancel the common factor.

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.4.2.3

Rewrite the expression.

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{\frac{2}{1}} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.1.4.2.4

Divide $2$ by $1$ .

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{2} - u_{1} + 6}} \cdot \frac{1}{2} d u_{1}$

Step 4.2

Multiply $\frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{2} - u_{1} + 6}}$ by $\frac{1}{2}$ .

$5 \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{2} - u_{1} + 6} \cdot 2} d u_{1}$

Step 4.3

Move $2$ to the left of $\sqrt{{u_{1}}^{2} - u_{1} + 6}$ .

$5 \int \frac{2 u_{1} - 1}{2 \sqrt{{u_{1}}^{2} - u_{1} + 6}} d u_{1}$

Step 5

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

$5 (\frac{1}{2} \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{2} - u_{1} + 6}} d u_{1})$

Step 6

Combine $\frac{1}{2}$ and $5$ .

$\frac{5}{2} \int \frac{2 u_{1} - 1}{\sqrt{{u_{1}}^{2} - u_{1} + 6}} d u_{1}$

Step 7

Let $u_{2} = {u_{1}}^{2} - u_{1} + 6$ . Then $d u_{2} = (2 u_{1} - 1) d u_{1}$ , so $\frac{1}{2 u_{1} - 1} d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 7.1

Let $u_{2} = {u_{1}}^{2} - u_{1} + 6$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 7.1.1

Differentiate ${u_{1}}^{2} - u_{1} + 6$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2} - u_{1} + 6]$

Step 7.1.2

Differentiate.

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Step 7.1.2.1

By the Sum Rule, the derivative of ${u_{1}}^{2} - u_{1} + 6$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [{u_{1}}^{2}] + \frac{d}{d u_{1}} [- u_{1}] + \frac{d}{d u_{1}} [6]$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] + \frac{d}{d u_{1}} [- u_{1}] + \frac{d}{d u_{1}} [6]$

Step 7.1.2.2

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

$2 u_{1} + \frac{d}{d u_{1}} [- u_{1}] + \frac{d}{d u_{1}} [6]$

Step 7.1.3

Evaluate $\frac{d}{d u_{1}} [- u_{1}]$ .

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Step 7.1.3.1

Since $- 1$ is constant with respect to $u_{1}$ , the derivative of $- u_{1}$ with respect to $u_{1}$ is $- \frac{d}{d u_{1}} [u_{1}]$ .

$2 u_{1} - \frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [6]$

Step 7.1.3.2

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

$2 u_{1} - 1 \cdot 1 + \frac{d}{d u_{1}} [6]$

Step 7.1.3.3

Multiply $- 1$ by $1$ .

$2 u_{1} - 1 + \frac{d}{d u_{1}} [6]$

Step 7.1.4

Differentiate using the Constant Rule.

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Step 7.1.4.1

Since $6$ is constant with respect to $u_{1}$ , the derivative of $6$ with respect to $u_{1}$ is $0$ .

$2 u_{1} - 1 + 0$

Step 7.1.4.2

Add $2 u_{1} - 1$ and $0$ .

$2 u_{1} - 1$

Step 7.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{5}{2} \int \frac{1}{\sqrt{u_{2}}} d u_{2}$

Step 8

Apply basic rules of exponents.

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Step 8.1

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

$\frac{5}{2} \int \frac{1}{{u_{2}}^{\frac{1}{2}}} d u_{2}$

Step 8.2

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

$\frac{5}{2} \int {({u_{2}}^{\frac{1}{2}})}^{- 1} d u_{2}$

Step 8.3

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

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Step 8.3.1

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

$\frac{5}{2} \int {u_{2}}^{\frac{1}{2} \cdot - 1} d u_{2}$

Step 8.3.2

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

$\frac{5}{2} \int {u_{2}}^{\frac{- 1}{2}} d u_{2}$

Step 8.3.3

Move the negative in front of the fraction.

$\frac{5}{2} \int {u_{2}}^{- \frac{1}{2}} d u_{2}$

Step 9

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

$\frac{5}{2} (2 {u_{2}}^{\frac{1}{2}} + C)$

Step 10

Simplify.

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Step 10.1

Rewrite $\frac{5}{2} (2 {u_{2}}^{\frac{1}{2}} + C)$ as $\frac{5}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C$ .

$\frac{5}{2} \cdot 2 {u_{2}}^{\frac{1}{2}} + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $\frac{5}{2}$ and $2$ .

$\frac{5 \cdot 2}{2} {u_{2}}^{\frac{1}{2}} + C$

Step 10.2.2

Multiply $5$ by $2$ .

$\frac{10}{2} {u_{2}}^{\frac{1}{2}} + C$

Step 10.2.3

Cancel the common factor of $10$ and $2$ .

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Step 10.2.3.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 5}{2} {u_{2}}^{\frac{1}{2}} + C$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 5}{2 (1)} {u_{2}}^{\frac{1}{2}} + C$

Step 10.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot 1} {u_{2}}^{\frac{1}{2}} + C$

Step 10.2.3.2.3

Rewrite the expression.

$\frac{5}{1} {u_{2}}^{\frac{1}{2}} + C$

Step 10.2.3.2.4

Divide $5$ by $1$ .

$5 {u_{2}}^{\frac{1}{2}} + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

Replace all occurrences of $u_{2}$ with ${u_{1}}^{2} - u_{1} + 6$ .

$5 {({u_{1}}^{2} - u_{1} + 6)}^{\frac{1}{2}} + C$

Step 11.2

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$5 {({(x^{2})}^{2} - x^{2} + 6)}^{\frac{1}{2}} + C$