Calculus Examples

$\int_{1}^{5} x \sqrt{2 x^{2} - 1} d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x^{2} - 1$ .

$\frac{d}{d x} [2 x^{2} - 1]$

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Multiply $2$ by $2$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$4 x + 0$

Step 1.1.4.2

Add $4 x$ and $0$ .

$4 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x^{2} - 1$ .

$u_{lower} = 2 \cdot 1^{2} - 1$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

One to any power is one.

$u_{lower} = 2 \cdot 1 - 1$

Step 1.3.1.2

Multiply $2$ by $1$ .

$u_{lower} = 2 - 1$

Step 1.3.2

Subtract $1$ from $2$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x^{2} - 1$ .

$u_{upper} = 2 \cdot 5^{2} - 1$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $5$ to the power of $2$ .

$u_{upper} = 2 \cdot 25 - 1$

Step 1.5.1.2

Multiply $2$ by $25$ .

$u_{upper} = 50 - 1$

Step 1.5.2

Subtract $1$ from $50$ .

$u_{upper} = 49$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 49$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{49} \sqrt{u} \frac{1}{4} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{4}$ .

$\int_{1}^{49} \frac{\sqrt{u}}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{1}^{49} \sqrt{u} d u$

Step 4

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

$\frac{1}{4} \int_{1}^{49} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{4} \frac{2}{3} u^{\frac{3}{2}}]_{1}^{49}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $49$ and at $1$ .

$\frac{1}{4} ((\frac{2}{3} \cdot 49^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $49$ as $7^{2}$ .

$\frac{1}{4} (\frac{2}{3} \cdot {(7^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{4} (\frac{2}{3} \cdot 7^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{4} (\frac{2}{3} \cdot 7^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{4} (\frac{2}{3} \cdot 7^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.4

Raise $7$ to the power of $3$ .

$\frac{1}{4} (\frac{2}{3} \cdot 343 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.5

Combine $\frac{2}{3}$ and $343$ .

$\frac{1}{4} (\frac{2 \cdot 343}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.6

Multiply $2$ by $343$ .

$\frac{1}{4} (\frac{686}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.7

One to any power is one.

$\frac{1}{4} (\frac{686}{3} - \frac{2}{3} \cdot 1)$

Step 6.2.8

Multiply $- 1$ by $1$ .

$\frac{1}{4} (\frac{686}{3} - \frac{2}{3})$

Step 6.2.9

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{686 - 2}{3}$

Step 6.2.10

Subtract $2$ from $686$ .

$\frac{1}{4} \cdot \frac{684}{3}$

Step 6.2.11

Cancel the common factor of $684$ and $3$ .

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Step 6.2.11.1

Factor $3$ out of $684$ .

$\frac{1}{4} \cdot \frac{3 \cdot 228}{3}$

Step 6.2.11.2

Cancel the common factors.

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Step 6.2.11.2.1

Factor $3$ out of $3$ .

$\frac{1}{4} \cdot \frac{3 \cdot 228}{3 (1)}$

Step 6.2.11.2.2

Cancel the common factor.

$\frac{1}{4} \cdot \frac{3 \cdot 228}{3 \cdot 1}$

Step 6.2.11.2.3

Rewrite the expression.

$\frac{1}{4} \cdot \frac{228}{1}$

Step 6.2.11.2.4

Divide $228$ by $1$ .

$\frac{1}{4} \cdot 228$

Step 6.2.12

Combine $\frac{1}{4}$ and $228$ .

$\frac{228}{4}$

Step 6.2.13

Cancel the common factor of $228$ and $4$ .

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Step 6.2.13.1

Factor $4$ out of $228$ .

$\frac{4 \cdot 57}{4}$

Step 6.2.13.2

Cancel the common factors.

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Step 6.2.13.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 57}{4 (1)}$

Step 6.2.13.2.2

Cancel the common factor.

$\frac{4 \cdot 57}{4 \cdot 1}$

Step 6.2.13.2.3

Rewrite the expression.

$\frac{57}{1}$

Step 6.2.13.2.4

Divide $57$ by $1$ .

$57$

Step 7