Calculus Examples

$\int_{3}^{4} x \sqrt{x - 3} d x$

Step 1

Let $u = x - 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x - 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 1.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 1.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 1.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $x$ in $u = x - 3$ .

$u_{lower} = 3 - 3$

Step 1.3

Subtract $3$ from $3$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = x - 3$ .

$u_{upper} = 4 - 3$

Step 1.5

Subtract $3$ from $4$ .

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 1.7

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

$\int_{0}^{1} (u + 3) \sqrt{u} d u$

Step 2

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Raise $u$ to the power of $1$ .

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

Step 3.3

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

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

Step 3.4

Write $1$ as a fraction with a common denominator.

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Add $2$ and $1$ .

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

Step 3.7

Reorder $u^{\frac{3}{2}}$ and $3 u^{\frac{1}{2}}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

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

Step 6

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

$3 (\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}) + \int_{0}^{1} u^{\frac{3}{2}} d u$

Step 7

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$3 (\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}) + \frac{2}{5} u^{\frac{5}{2}}]_{0}^{1}$

Step 8

Simplify the expression.

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Step 8.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $1$ and at $0$ .

$3 ((\frac{2}{3} \cdot 1^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}}) + \frac{2}{5} u^{\frac{5}{2}}]_{0}^{1}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Evaluate $\frac{2}{5} u^{\frac{5}{2}}$ at $1$ and at $0$ .

$3 (\frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.2.2

One to any power is one.

$3 (\frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0^{\frac{3}{2}}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.2.3

Multiply $\frac{2}{3}$ by $1$ .

$3 (\frac{2}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.3

Simplify.

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Step 8.3.1

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

$3 (\frac{2}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.3.2

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

$3 (\frac{2}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.3.3

Cancel the common factor of $2$ .

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Step 8.3.3.1

Cancel the common factor.

$3 (\frac{2}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.3.3.2

Rewrite the expression.

$3 (\frac{2}{3} - \frac{2}{3} \cdot 0^{3}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.3.4

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

$3 (\frac{2}{3} - \frac{2}{3} \cdot 0) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.4

Simplify.

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Step 8.4.1

Multiply $0$ by $- 1$ .

$3 (\frac{2}{3} + 0 (\frac{2}{3})) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.4.2

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

$3 (\frac{2}{3} + 0) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.5

Add $\frac{2}{3}$ and $0$ .

$3 (\frac{2}{3}) + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6

Simplify.

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Step 8.6.1

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

$\frac{3 \cdot 2}{3} + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6.2

Multiply $3$ by $2$ .

$\frac{6}{3} + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6.3

Cancel the common factor of $6$ and $3$ .

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Step 8.6.3.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 2}{3} + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6.3.2

Cancel the common factors.

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Step 8.6.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 2}{3 (1)} + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6.3.2.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 \cdot 1} + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6.3.2.3

Rewrite the expression.

$\frac{2}{1} + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.6.3.2.4

Divide $2$ by $1$ .

$2 + \frac{2}{5} \cdot 1^{\frac{5}{2}} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.7

Simplify the expression.

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Step 8.7.1

One to any power is one.

$2 + \frac{2}{5} \cdot 1 - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.7.2

Multiply $\frac{2}{5}$ by $1$ .

$2 + \frac{2}{5} - \frac{2}{5} \cdot 0^{\frac{5}{2}}$

Step 8.8

Simplify.

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Step 8.8.1

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

$2 + \frac{2}{5} - \frac{2}{5} \cdot {(0^{2})}^{\frac{5}{2}}$

Step 8.8.2

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

$2 + \frac{2}{5} - \frac{2}{5} \cdot 0^{2 (\frac{5}{2})}$

Step 8.8.3

Cancel the common factor of $2$ .

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Step 8.8.3.1

Cancel the common factor.

$2 + \frac{2}{5} - \frac{2}{5} \cdot 0^{2 (\frac{5}{2})}$

Step 8.8.3.2

Rewrite the expression.

$2 + \frac{2}{5} - \frac{2}{5} \cdot 0^{5}$

Step 8.8.4

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

$2 + \frac{2}{5} - \frac{2}{5} \cdot 0$

Step 8.9

Simplify.

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Step 8.9.1

Multiply $0$ by $- 1$ .

$2 + \frac{2}{5} + 0 (\frac{2}{5})$

Step 8.9.2

Multiply $0$ by $\frac{2}{5}$ .

$2 + \frac{2}{5} + 0$

Step 8.10

Add $\frac{2}{5}$ and $0$ .

$2 + \frac{2}{5}$

Step 8.11

Simplify.

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Step 8.11.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$2 \cdot \frac{5}{5} + \frac{2}{5}$

Step 8.11.2

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

$\frac{2 \cdot 5}{5} + \frac{2}{5}$

Step 8.11.3

Combine the numerators over the common denominator.

$\frac{2 \cdot 5 + 2}{5}$

Step 8.11.4

Simplify the numerator.

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Step 8.11.4.1

Multiply $2$ by $5$ .

$\frac{10 + 2}{5}$

Step 8.11.4.2

Add $10$ and $2$ .

$\frac{12}{5}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{5}$

Decimal Form:

$2.4$

Mixed Number Form:

$2 \frac{2}{5}$