Calculus Examples

$\int_{0}^{1} x^{2} \sqrt{3 x^{3} + 6} d x$

Step 1

Let $u = 3 x^{3} + 6$ . Then $d u = 9 x^{2} d x$ , so $\frac{1}{9} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x^{3} + 6$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x^{3} + 6$ .

$\frac{d}{d x} [3 x^{3} + 6]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x^{3} + 6$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [6]$

Step 1.1.3

Evaluate $\frac{d}{d x} [3 x^{3}]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x^{3}]$ .

$3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6]$

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 $n = 3$ .

$3 (3 x^{2}) + \frac{d}{d x} [6]$

Step 1.1.3.3

Multiply $3$ by $3$ .

$9 x^{2} + \frac{d}{d x} [6]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$9 x^{2} + 0$

Step 1.1.4.2

Add $9 x^{2}$ and $0$ .

$9 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 x^{3} + 6$ .

$u_{lower} = 3 \cdot 0^{3} + 6$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$u_{lower} = 3 \cdot 0 + 6$

Step 1.3.1.2

Multiply $3$ by $0$ .

$u_{lower} = 0 + 6$

Step 1.3.2

Add $0$ and $6$ .

$u_{lower} = 6$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 x^{3} + 6$ .

$u_{upper} = 3 \cdot 1^{3} + 6$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 3 \cdot 1 + 6$

Step 1.5.1.2

Multiply $3$ by $1$ .

$u_{upper} = 3 + 6$

Step 1.5.2

Add $3$ and $6$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 6$

$u_{upper} = 9$

Step 1.7

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

$\int_{6}^{9} \sqrt{u} \frac{1}{9} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{9}$ .

$\int_{6}^{9} \frac{\sqrt{u}}{9} d u$

Step 3

Since $\frac{1}{9}$ is constant with respect to $u$ , move $\frac{1}{9}$ out of the integral.

$\frac{1}{9} \int_{6}^{9} \sqrt{u} d u$

Step 4

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

$\frac{1}{9} \int_{6}^{9} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{9} \frac{2}{3} u^{\frac{3}{2}}]_{6}^{9}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $6$ .

$\frac{1}{9} ((\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{1}{9} (\frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{9} (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{9} (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{9} (\frac{2}{3} \cdot 3^{3} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.4

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

$\frac{1}{9} (\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.5

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

$\frac{1}{9} (\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.6

Multiply $2$ by $27$ .

$\frac{1}{9} (\frac{54}{3} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.7

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

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Step 6.2.7.1

Factor $3$ out of $54$ .

$\frac{1}{9} (\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.7.2

Cancel the common factors.

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Step 6.2.7.2.1

Factor $3$ out of $3$ .

$\frac{1}{9} (\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.7.2.2

Cancel the common factor.

$\frac{1}{9} (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.7.2.3

Rewrite the expression.

$\frac{1}{9} (\frac{18}{1} - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.7.2.4

Divide $18$ by $1$ .

$\frac{1}{9} (18 - \frac{2}{3} \cdot 6^{\frac{3}{2}})$

Step 6.2.8

Combine $6^{\frac{3}{2}}$ and $\frac{2}{3}$ .

$\frac{1}{9} (18 - \frac{6^{\frac{3}{2}} \cdot 2}{3})$

Step 6.2.9

Move $2$ to the left of $6^{\frac{3}{2}}$ .

$\frac{1}{9} (18 - \frac{2 \cdot 6^{\frac{3}{2}}}{3})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{9} \cdot 18 + \frac{1}{9} (- \frac{2 \cdot 6^{\frac{3}{2}}}{3})$

Step 7.2

Cancel the common factor of $9$ .

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Step 7.2.1

Factor $9$ out of $18$ .

$\frac{1}{9} \cdot (9 (2)) + \frac{1}{9} (- \frac{2 \cdot 6^{\frac{3}{2}}}{3})$

Step 7.2.2

Cancel the common factor.

$\frac{1}{9} \cdot (9 \cdot 2) + \frac{1}{9} (- \frac{2 \cdot 6^{\frac{3}{2}}}{3})$

Step 7.2.3

Rewrite the expression.

$2 + \frac{1}{9} (- \frac{2 \cdot 6^{\frac{3}{2}}}{3})$

Step 7.3

Multiply $\frac{1}{9} (- \frac{2 \cdot 6^{\frac{3}{2}}}{3})$ .

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Step 7.3.1

Multiply $\frac{1}{9}$ by $\frac{2 \cdot 6^{\frac{3}{2}}}{3}$ .

$2 - \frac{2 \cdot 6^{\frac{3}{2}}}{9 \cdot 3}$

Step 7.3.2

Multiply $9$ by $3$ .

$2 - \frac{2 \cdot 6^{\frac{3}{2}}}{27}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$2 - \frac{2 \cdot 6^{\frac{3}{2}}}{27}$

Decimal Form:

$0.91133789 \dots$

Step 9