Calculus Examples

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

Step 1

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

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Step 1.1.1

Differentiate $x^{2} - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

$u_{lower} = 1 - 1$

Step 1.3.2

Subtract $1$ from $1$ .

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = 2^{2} - 1$

Step 1.5

Simplify.

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Step 1.5.1

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

$u_{upper} = 4 - 1$

Step 1.5.2

Subtract $1$ from $4$ .

$u_{upper} = 3$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 3$

Step 1.7

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

$\int_{0}^{3} \sqrt{u} \frac{1}{2} d u$

Step 2

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

$\int_{0}^{3} \frac{\sqrt{u}}{2} d u$

Step 3

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

$\frac{1}{2} \int_{0}^{3} \sqrt{u} d u$

Step 4

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

$\frac{1}{2} \int_{0}^{3} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{2} \frac{2}{3} u^{\frac{3}{2}}]_{0}^{3}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $3$ and at $0$ .

$\frac{1}{2} ((\frac{2}{3} \cdot 3^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{1}{2} (\frac{2 \cdot 3^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\frac{1}{2} (2 \cdot 3^{\frac{3}{2}} \cdot 3^{- 1} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3

Multiply $3^{\frac{3}{2}}$ by $3^{- 1}$ by adding the exponents.

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Step 6.2.3.1

Move $3^{- 1}$ .

$\frac{1}{2} (2 \cdot (3^{- 1} \cdot 3^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.2

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

$\frac{1}{2} (2 \cdot 3^{- 1 + \frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (2 \cdot 3^{- 1 \cdot \frac{2}{2} + \frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.4

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

$\frac{1}{2} (2 \cdot 3^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.5

Combine the numerators over the common denominator.

$\frac{1}{2} (2 \cdot 3^{\frac{- 1 \cdot 2 + 3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.6

Simplify the numerator.

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Step 6.2.3.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} (2 \cdot 3^{\frac{- 2 + 3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.6.2

Add $- 2$ and $3$ .

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.4

Rewrite $0$ as $0^{2}$ .

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}})$

Step 6.2.5

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

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.6

Cancel the common factor of $2$ .

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Step 6.2.6.1

Cancel the common factor.

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.6.2

Rewrite the expression.

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} - \frac{2}{3} \cdot 0^{3})$

Step 6.2.7

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} - \frac{2}{3} \cdot 0)$

Step 6.2.8

Multiply $0$ by $- 1$ .

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} + 0 (\frac{2}{3}))$

Step 6.2.9

Multiply $0$ by $\frac{2}{3}$ .

$\frac{1}{2} (2 \cdot 3^{\frac{1}{2}} + 0)$

Step 6.2.10

Add $2 \cdot 3^{\frac{1}{2}}$ and $0$ .

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

Step 6.2.11

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

$\frac{2}{2} \cdot 3^{\frac{1}{2}}$

Step 6.2.12

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

$\frac{2 \cdot 3^{\frac{1}{2}}}{2}$

Step 6.2.13

Cancel the common factor.

$\frac{2 \cdot 3^{\frac{1}{2}}}{2}$

Step 6.2.14

Divide $3^{\frac{1}{2}}$ by $1$ .

$3^{\frac{1}{2}}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$3^{\frac{1}{2}}$

Decimal Form:

$1.73205080 \dots$

Step 8