Calculus Examples

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

Step 1

Remove parentheses.

$\int_{0}^{4} 1 + 3 x - x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{4} d x + \int_{0}^{4} 3 x d x + \int_{0}^{4} - x^{2} d x$

Step 3

Apply the constant rule.

$x]_{0}^{4} + \int_{0}^{4} 3 x d x + \int_{0}^{4} - x^{2} d x$

Step 4

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

$x]_{0}^{4} + 3 \int_{0}^{4} x d x + \int_{0}^{4} - x^{2} d x$

Step 5

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

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

Step 6

Combine $\frac{1}{2}$ and $x^{2}$ .

$x]_{0}^{4} + 3 (\frac{x^{2}}{2}]_{0}^{4}) + \int_{0}^{4} - x^{2} d x$

Step 7

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

$x]_{0}^{4} + 3 (\frac{x^{2}}{2}]_{0}^{4}) - \int_{0}^{4} x^{2} d x$

Step 8

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

$x]_{0}^{4} + 3 (\frac{x^{2}}{2}]_{0}^{4}) - (\frac{1}{3} x^{3}]_{0}^{4})$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$x]_{0}^{4} + 3 (\frac{x^{2}}{2}]_{0}^{4}) - (\frac{x^{3}}{3}]_{0}^{4})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $x$ at $4$ and at $0$ .

$(4) + 0 + 3 (\frac{x^{2}}{2}]_{0}^{4}) - (\frac{x^{3}}{3}]_{0}^{4})$

Step 9.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $0$ .

$4 + 0 + 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2}) - (\frac{x^{3}}{3}]_{0}^{4})$

Step 9.2.3

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

$4 + 0 + 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4

Simplify.

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Step 9.2.4.1

Add $4$ and $0$ .

$4 + 3 (\frac{4^{2}}{2} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.2

Raise $4$ to the power of $2$ .

$4 + 3 (\frac{16}{2} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.3

Cancel the common factor of $16$ and $2$ .

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Step 9.2.4.3.1

Factor $2$ out of $16$ .

$4 + 3 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.3.2

Cancel the common factors.

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Step 9.2.4.3.2.1

Factor $2$ out of $2$ .

$4 + 3 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.3.2.2

Cancel the common factor.

$4 + 3 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.3.2.3

Rewrite the expression.

$4 + 3 (\frac{8}{1} - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.3.2.4

Divide $8$ by $1$ .

$4 + 3 (8 - \frac{0^{2}}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.4

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

$4 + 3 (8 - \frac{0}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.5

Cancel the common factor of $0$ and $2$ .

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Step 9.2.4.5.1

Factor $2$ out of $0$ .

$4 + 3 (8 - \frac{2 (0)}{2}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.5.2

Cancel the common factors.

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Step 9.2.4.5.2.1

Factor $2$ out of $2$ .

$4 + 3 (8 - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.5.2.2

Cancel the common factor.

$4 + 3 (8 - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.5.2.3

Rewrite the expression.

$4 + 3 (8 - \frac{0}{1}) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.5.2.4

Divide $0$ by $1$ .

$4 + 3 (8 - 0) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.6

Multiply $- 1$ by $0$ .

$4 + 3 (8 + 0) - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.7

Add $8$ and $0$ .

$4 + 3 \cdot 8 - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.8

Multiply $3$ by $8$ .

$4 + 24 - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.9

Add $4$ and $24$ .

$28 - (\frac{4^{3}}{3} - \frac{0^{3}}{3})$

Step 9.2.4.10

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

$28 - (\frac{64}{3} - \frac{0^{3}}{3})$

Step 9.2.4.11

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

$28 - (\frac{64}{3} - \frac{0}{3})$

Step 9.2.4.12

Cancel the common factor of $0$ and $3$ .

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Step 9.2.4.12.1

Factor $3$ out of $0$ .

$28 - (\frac{64}{3} - \frac{3 (0)}{3})$

Step 9.2.4.12.2

Cancel the common factors.

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Step 9.2.4.12.2.1

Factor $3$ out of $3$ .

$28 - (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 9.2.4.12.2.2

Cancel the common factor.

$28 - (\frac{64}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 9.2.4.12.2.3

Rewrite the expression.

$28 - (\frac{64}{3} - \frac{0}{1})$

Step 9.2.4.12.2.4

Divide $0$ by $1$ .

$28 - (\frac{64}{3} - 0)$

Step 9.2.4.13

Multiply $- 1$ by $0$ .

$28 - (\frac{64}{3} + 0)$

Step 9.2.4.14

Add $\frac{64}{3}$ and $0$ .

$28 - \frac{64}{3}$

Step 9.2.4.15

To write $28$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$28 \cdot \frac{3}{3} - \frac{64}{3}$

Step 9.2.4.16

Combine $28$ and $\frac{3}{3}$ .

$\frac{28 \cdot 3}{3} - \frac{64}{3}$

Step 9.2.4.17

Combine the numerators over the common denominator.

$\frac{28 \cdot 3 - 64}{3}$

Step 9.2.4.18

Simplify the numerator.

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Step 9.2.4.18.1

Multiply $28$ by $3$ .

$\frac{84 - 64}{3}$

Step 9.2.4.18.2

Subtract $64$ from $84$ .

$\frac{20}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{20}{3}$

Decimal Form:

$6.\overline{6}$

Mixed Number Form:

$6 \frac{2}{3}$

Step 11