Calculus Examples

$\int_{0}^{1} (x^{3} + x) {(x^{4} + 2 x^{2} + 9)}^{\frac{1}{2}} d x$

Step 1

Let $u = x^{4} + 2 x^{2} + 9$ . Then $d u = (4 x^{3} + 4 x) d x$ , so $\frac{1}{4} d u = (x^{3} + x) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{4} + 2 x^{2} + 9$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{4} + 2 x^{2} + 9$ .

$\frac{d}{d x} [x^{4} + 2 x^{2} + 9]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{4} + 2 x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [9]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [9]$

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

$4 x^{3} + 2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

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 $n = 2$ .

$4 x^{3} + 2 (2 x) + \frac{d}{d x} [9]$

Step 1.1.3.3

Multiply $2$ by $2$ .

$4 x^{3} + 4 x + \frac{d}{d x} [9]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$4 x^{3} + 4 x + 0$

Step 1.1.4.2

Add $4 x^{3} + 4 x$ and $0$ .

$4 x^{3} + 4 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{4} + 2 x^{2} + 9$ .

$u_{lower} = 0^{4} + 2 \cdot 0^{2} + 9$

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 $0$ .

$u_{lower} = 0 + 2 \cdot 0^{2} + 9$

Step 1.3.1.2

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

$u_{lower} = 0 + 2 \cdot 0 + 9$

Step 1.3.1.3

Multiply $2$ by $0$ .

$u_{lower} = 0 + 0 + 9$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0 + 9$

Step 1.3.3

Add $0$ and $9$ .

$u_{lower} = 9$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{4} + 2 x^{2} + 9$ .

$u_{upper} = 1^{4} + 2 \cdot 1^{2} + 9$

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.

$u_{upper} = 1 + 2 \cdot 1^{2} + 9$

Step 1.5.1.2

One to any power is one.

$u_{upper} = 1 + 2 \cdot 1 + 9$

Step 1.5.1.3

Multiply $2$ by $1$ .

$u_{upper} = 1 + 2 + 9$

Step 1.5.2

Add $1$ and $2$ .

$u_{upper} = 3 + 9$

Step 1.5.3

Add $3$ and $9$ .

$u_{upper} = 12$

Step 1.6

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

$u_{lower} = 9$

$u_{upper} = 12$

Step 1.7

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

$\int_{9}^{12} u^{\frac{1}{2}} \frac{1}{4} d u$

Step 2

Combine $u^{\frac{1}{2}}$ and $\frac{1}{4}$ .

$\int_{9}^{12} \frac{u^{\frac{1}{2}}}{4} d u$

Step 3

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

$\frac{1}{4} \int_{9}^{12} u^{\frac{1}{2}} d u$

Step 4

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

$\frac{1}{4} \frac{2}{3} u^{\frac{3}{2}}]_{9}^{12}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $12$ and at $9$ .

$\frac{1}{4} ((\frac{2}{3} \cdot 12^{\frac{3}{2}}) - \frac{2}{3} \cdot 9^{\frac{3}{2}})$

Step 5.2

Simplify.

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Step 5.2.1

Combine $\frac{2}{3}$ and $12^{\frac{3}{2}}$ .

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 9^{\frac{3}{2}})$

Step 5.2.2

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

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}})$

Step 5.2.3

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

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 3^{2 (\frac{3}{2})})$

Step 5.2.4

Cancel the common factor of $2$ .

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Step 5.2.4.1

Cancel the common factor.

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 3^{2 (\frac{3}{2})})$

Step 5.2.4.2

Rewrite the expression.

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 3^{3})$

Step 5.2.5

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

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 27)$

Step 5.2.6

Multiply $27$ by $- 1$ .

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - 27 (\frac{2}{3}))$

Step 5.2.7

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

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{- 27 \cdot 2}{3})$

Step 5.2.8

Multiply $- 27$ by $2$ .

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{- 54}{3})$

Step 5.2.9

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

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Step 5.2.9.1

Factor $3$ out of $- 54$ .

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{3 \cdot - 18}{3})$

Step 5.2.9.2

Cancel the common factors.

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Step 5.2.9.2.1

Factor $3$ out of $3$ .

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{3 \cdot - 18}{3 (1)})$

Step 5.2.9.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{3 \cdot - 18}{3 \cdot 1})$

Step 5.2.9.2.3

Rewrite the expression.

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{- 18}{1})$

Step 5.2.9.2.4

Divide $- 18$ by $1$ .

$\frac{1}{4} (\frac{2 \cdot 12^{\frac{3}{2}}}{3} - 18)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{1}{4} \cdot \frac{2 \cdot 12^{\frac{3}{2}}}{3} + \frac{1}{4} \cdot - 18$

Step 6.2

Combine.

$\frac{1 (2 \cdot 12^{\frac{3}{2}})}{4 \cdot 3} + \frac{1}{4} \cdot - 18$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Factor $2$ out of $4$ .

$\frac{1 (2 \cdot 12^{\frac{3}{2}})}{4 \cdot 3} + \frac{1}{2 (2)} \cdot - 18$

Step 6.3.2

Factor $2$ out of $- 18$ .

$\frac{1 (2 \cdot 12^{\frac{3}{2}})}{4 \cdot 3} + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 9)$

Step 6.3.3

Cancel the common factor.

$\frac{1 (2 \cdot 12^{\frac{3}{2}})}{4 \cdot 3} + \frac{1}{2 \cdot 2} \cdot (2 \cdot - 9)$

Step 6.3.4

Rewrite the expression.

$\frac{1 (2 \cdot 12^{\frac{3}{2}})}{4 \cdot 3} + \frac{1}{2} \cdot - 9$

Step 6.4

Combine $\frac{1}{2}$ and $- 9$ .

$\frac{1 (2 \cdot 12^{\frac{3}{2}})}{4 \cdot 3} + \frac{- 9}{2}$

Step 6.5

Simplify each term.

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Step 6.5.1

Factor $2$ out of $1 (2 \cdot 12^{\frac{3}{2}})$ .

$\frac{2 (1 \cdot (12^{\frac{3}{2}}))}{4 \cdot 3} + \frac{- 9}{2}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor $2$ out of $4 \cdot 3$ .

$\frac{2 (1 \cdot (12^{\frac{3}{2}}))}{2 (2 \cdot 3)} + \frac{- 9}{2}$

Step 6.5.2.2

Cancel the common factor.

$\frac{2 (1 \cdot (12^{\frac{3}{2}}))}{2 (2 \cdot 3)} + \frac{- 9}{2}$

Step 6.5.2.3

Rewrite the expression.

$\frac{1 \cdot (12^{\frac{3}{2}})}{2 \cdot 3} + \frac{- 9}{2}$

Step 6.5.3

Multiply $12^{\frac{3}{2}}$ by $1$ .

$\frac{12^{\frac{3}{2}}}{2 \cdot 3} + \frac{- 9}{2}$

Step 6.5.4

Multiply $2$ by $3$ .

$\frac{12^{\frac{3}{2}}}{6} + \frac{- 9}{2}$

Step 6.5.5

Move the negative in front of the fraction.

$\frac{12^{\frac{3}{2}}}{6} - \frac{9}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{12^{\frac{3}{2}}}{6} - \frac{9}{2}$

Decimal Form:

$2.42820323 \dots$

Step 8