Calculus Examples

$\int_{- 2}^{- 1} 2 x {(4 - x^{2})}^{3} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{- 2}^{- 1} x {(4 - x^{2})}^{3} d x$

Step 2

Let $u = 4 - x^{2}$ . 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 2.1

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

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Step 2.1.1

Differentiate $4 - x^{2}$ .

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 2.1.3

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

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Step 2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 2.1.3.2

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

$0 - (2 x)$

Step 2.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 2.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 2.2

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

$u_{lower} = 4 - {(- 2)}^{2}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

$u_{lower} = 4 - 1 \cdot 4$

Step 2.3.1.2

Multiply $- 1$ by $4$ .

$u_{lower} = 4 - 4$

Step 2.3.2

Subtract $4$ from $4$ .

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = 4 - {(- 1)}^{2}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.5.1.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 2.5.1.1.1.1

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

$u_{upper} = 4 + {(- 1)}^{1} {(- 1)}^{2}$

Step 2.5.1.1.1.2

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

$u_{upper} = 4 + {(- 1)}^{1 + 2}$

Step 2.5.1.1.2

Add $1$ and $2$ .

$u_{upper} = 4 + {(- 1)}^{3}$

Step 2.5.1.2

Raise $- 1$ to the power of $3$ .

$u_{upper} = 4 - 1$

Step 2.5.2

Subtract $1$ from $4$ .

$u_{upper} = 3$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 3$

Step 2.7

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

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

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

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

Step 3.2

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

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

Step 4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 5

Multiply $- 1$ by $2$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Cancel the common factor of $- 2$ and $2$ .

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Step 7.2.1

Factor $2$ out of $- 2$ .

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

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $2$ out of $2$ .

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

Step 7.2.2.2

Cancel the common factor.

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

Step 7.2.2.3

Rewrite the expression.

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

Step 7.2.2.4

Divide $- 1$ by $1$ .

$- \int_{0}^{3} u^{3} d u$

Step 8

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$- (\frac{1}{4} u^{4}]_{0}^{3})$

Step 9

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

$- (\frac{u^{4}}{4}]_{0}^{3})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{u^{4}}{4}$ at $3$ and at $0$ .

$- ((\frac{3^{4}}{4}) - \frac{0^{4}}{4})$

Step 10.2

Simplify.

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Step 10.2.1

Raise $3$ to the power of $4$ .

$- (\frac{81}{4} - \frac{0^{4}}{4})$

Step 10.2.2

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

$- (\frac{81}{4} - \frac{0}{4})$

Step 10.2.3

Cancel the common factor of $0$ and $4$ .

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Step 10.2.3.1

Factor $4$ out of $0$ .

$- (\frac{81}{4} - \frac{4 (0)}{4})$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $4$ out of $4$ .

$- (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 10.2.3.2.2

Cancel the common factor.

$- (\frac{81}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 10.2.3.2.3

Rewrite the expression.

$- (\frac{81}{4} - \frac{0}{1})$

Step 10.2.3.2.4

Divide $0$ by $1$ .

$- (\frac{81}{4} - 0)$

Step 10.2.4

Multiply $- 1$ by $0$ .

$- (\frac{81}{4} + 0)$

Step 10.2.5

Add $\frac{81}{4}$ and $0$ .

$- \frac{81}{4}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{81}{4}$

Decimal Form:

$- 20.25$

Mixed Number Form:

$- 20 \frac{1}{4}$

Step 12