Calculus Examples

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

Step 1

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

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Step 1.1

Let $u = x^{3} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} + 1$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [1]$ .

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [1]$

Step 1.1.4

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

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{3} + 1$ .

$u_{lower} = 0^{3} + 1$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{3} + 1$ .

$u_{upper} = 2^{3} + 1$

Step 1.5

Simplify.

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Step 1.5.1

Raise $2$ to the power of $3$ .

$u_{upper} = 8 + 1$

Step 1.5.2

Add $8$ and $1$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 1.7

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

$\int_{1}^{9} u^{\frac{1}{2}} \frac{1}{3} d u$

Step 2

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

$\int_{1}^{9} \frac{u^{\frac{1}{2}}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{1}^{9} u^{\frac{1}{2}} d u$

Step 4

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

$\frac{1}{3} \frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $1$ .

$\frac{1}{3} ((\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2

Simplify.

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Step 5.2.1

Rewrite $9$ as $3^{2}$ .

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor of $2$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Rewrite the expression.

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

Step 5.2.4

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

$\frac{1}{3} (\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.5

Combine $\frac{2}{3}$ and $27$ .

$\frac{1}{3} (\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.6

Multiply $2$ by $27$ .

$\frac{1}{3} (\frac{54}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.7

Cancel the common factor of $54$ and $3$ .

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Step 5.2.7.1

Factor $3$ out of $54$ .

$\frac{1}{3} (\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.7.2

Cancel the common factors.

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Step 5.2.7.2.1

Factor $3$ out of $3$ .

$\frac{1}{3} (\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.7.2.2

Cancel the common factor.

$\frac{1}{3} (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.7.2.3

Rewrite the expression.

$\frac{1}{3} (\frac{18}{1} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.7.2.4

Divide $18$ by $1$ .

$\frac{1}{3} (18 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.8

One to any power is one.

$\frac{1}{3} (18 - \frac{2}{3} \cdot 1)$

Step 5.2.9

Multiply $- 1$ by $1$ .

$\frac{1}{3} (18 - \frac{2}{3})$

Step 5.2.10

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} (18 \cdot \frac{3}{3} - \frac{2}{3})$

Step 5.2.11

Combine $18$ and $\frac{3}{3}$ .

$\frac{1}{3} (\frac{18 \cdot 3}{3} - \frac{2}{3})$

Step 5.2.12

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{18 \cdot 3 - 2}{3}$

Step 5.2.13

Simplify the numerator.

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Step 5.2.13.1

Multiply $18$ by $3$ .

$\frac{1}{3} \cdot \frac{54 - 2}{3}$

Step 5.2.13.2

Subtract $2$ from $54$ .

$\frac{1}{3} \cdot \frac{52}{3}$

Step 5.2.14

Multiply $\frac{1}{3}$ by $\frac{52}{3}$ .

$\frac{52}{3 \cdot 3}$

Step 5.2.15

Multiply $3$ by $3$ .

$\frac{52}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{52}{9}$

Decimal Form:

$5.\overline{7}$

Mixed Number Form:

$5 \frac{7}{9}$

Step 7