Calculus Examples

$\int_{0}^{1} (1 + 4 x + 9 x^{2} + 16 x^{3}) d x$

Step 1

Remove parentheses.

$\int_{0}^{1} 1 + 4 x + 9 x^{2} + 16 x^{3} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} d x + \int_{0}^{1} 4 x d x + \int_{0}^{1} 9 x^{2} d x + \int_{0}^{1} 16 x^{3} d x$

Step 3

Apply the constant rule.

$x]_{0}^{1} + \int_{0}^{1} 4 x d x + \int_{0}^{1} 9 x^{2} d x + \int_{0}^{1} 16 x^{3} d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$x]_{0}^{1} + 4 \int_{0}^{1} x d x + \int_{0}^{1} 9 x^{2} d x + \int_{0}^{1} 16 x^{3} d x$

Step 5

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

$x]_{0}^{1} + 4 (\frac{1}{2} x^{2}]_{0}^{1}) + \int_{0}^{1} 9 x^{2} d x + \int_{0}^{1} 16 x^{3} d x$

Step 6

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

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + \int_{0}^{1} 9 x^{2} d x + \int_{0}^{1} 16 x^{3} d x$

Step 7

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 \int_{0}^{1} x^{2} d x + \int_{0}^{1} 16 x^{3} d x$

Step 8

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

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} 16 x^{3} d x$

Step 9

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

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} 16 x^{3} d x$

Step 10

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 (\frac{x^{3}}{3}]_{0}^{1}) + 16 \int_{0}^{1} x^{3} d x$

Step 11

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

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 (\frac{x^{3}}{3}]_{0}^{1}) + 16 (\frac{1}{4} x^{4}]_{0}^{1})$

Step 12

Simplify the answer.

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Step 12.1

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

$x]_{0}^{1} + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 (\frac{x^{3}}{3}]_{0}^{1}) + 16 (\frac{x^{4}}{4}]_{0}^{1})$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate $x$ at $1$ and at $0$ .

$(1) + 0 + 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 (\frac{x^{3}}{3}]_{0}^{1}) + 16 (\frac{x^{4}}{4}]_{0}^{1})$

Step 12.2.2

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $0$ .

$1 + 0 + 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 9 (\frac{x^{3}}{3}]_{0}^{1}) + 16 (\frac{x^{4}}{4}]_{0}^{1})$

Step 12.2.3

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

$1 + 0 + 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{x^{4}}{4}]_{0}^{1})$

Step 12.2.4

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

$1 + 0 + 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5

Simplify.

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Step 12.2.5.1

Add $1$ and $0$ .

$1 + 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.2

One to any power is one.

$1 + 4 (\frac{1}{2} - \frac{0^{2}}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.3

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

$1 + 4 (\frac{1}{2} - \frac{0}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.4

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

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Step 12.2.5.4.1

Factor $2$ out of $0$ .

$1 + 4 (\frac{1}{2} - \frac{2 (0)}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.4.2

Cancel the common factors.

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Step 12.2.5.4.2.1

Factor $2$ out of $2$ .

$1 + 4 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.4.2.2

Cancel the common factor.

$1 + 4 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.4.2.3

Rewrite the expression.

$1 + 4 (\frac{1}{2} - \frac{0}{1}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.4.2.4

Divide $0$ by $1$ .

$1 + 4 (\frac{1}{2} - 0) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.5

Multiply $- 1$ by $0$ .

$1 + 4 (\frac{1}{2} + 0) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.6

Add $\frac{1}{2}$ and $0$ .

$1 + 4 (\frac{1}{2}) + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.7

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

$1 + \frac{4}{2} + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.8

Cancel the common factor of $4$ and $2$ .

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Step 12.2.5.8.1

Factor $2$ out of $4$ .

$1 + \frac{2 \cdot 2}{2} + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.8.2

Cancel the common factors.

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Step 12.2.5.8.2.1

Factor $2$ out of $2$ .

$1 + \frac{2 \cdot 2}{2 (1)} + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.8.2.2

Cancel the common factor.

$1 + \frac{2 \cdot 2}{2 \cdot 1} + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.8.2.3

Rewrite the expression.

$1 + \frac{2}{1} + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.8.2.4

Divide $2$ by $1$ .

$1 + 2 + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.9

Add $1$ and $2$ .

$3 + 9 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.10

One to any power is one.

$3 + 9 (\frac{1}{3} - \frac{0^{3}}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.11

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

$3 + 9 (\frac{1}{3} - \frac{0}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.12

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

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Step 12.2.5.12.1

Factor $3$ out of $0$ .

$3 + 9 (\frac{1}{3} - \frac{3 (0)}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.12.2

Cancel the common factors.

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Step 12.2.5.12.2.1

Factor $3$ out of $3$ .

$3 + 9 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.12.2.2

Cancel the common factor.

$3 + 9 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.12.2.3

Rewrite the expression.

$3 + 9 (\frac{1}{3} - \frac{0}{1}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.12.2.4

Divide $0$ by $1$ .

$3 + 9 (\frac{1}{3} - 0) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.13

Multiply $- 1$ by $0$ .

$3 + 9 (\frac{1}{3} + 0) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.14

Add $\frac{1}{3}$ and $0$ .

$3 + 9 (\frac{1}{3}) + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.15

Combine $9$ and $\frac{1}{3}$ .

$3 + \frac{9}{3} + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.16

Cancel the common factor of $9$ and $3$ .

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Step 12.2.5.16.1

Factor $3$ out of $9$ .

$3 + \frac{3 \cdot 3}{3} + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.16.2

Cancel the common factors.

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Step 12.2.5.16.2.1

Factor $3$ out of $3$ .

$3 + \frac{3 \cdot 3}{3 (1)} + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.16.2.2

Cancel the common factor.

$3 + \frac{3 \cdot 3}{3 \cdot 1} + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.16.2.3

Rewrite the expression.

$3 + \frac{3}{1} + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.16.2.4

Divide $3$ by $1$ .

$3 + 3 + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.17

Add $3$ and $3$ .

$6 + 16 (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 12.2.5.18

One to any power is one.

$6 + 16 (\frac{1}{4} - \frac{0^{4}}{4})$

Step 12.2.5.19

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

$6 + 16 (\frac{1}{4} - \frac{0}{4})$

Step 12.2.5.20

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

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Step 12.2.5.20.1

Factor $4$ out of $0$ .

$6 + 16 (\frac{1}{4} - \frac{4 (0)}{4})$

Step 12.2.5.20.2

Cancel the common factors.

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Step 12.2.5.20.2.1

Factor $4$ out of $4$ .

$6 + 16 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 12.2.5.20.2.2

Cancel the common factor.

$6 + 16 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 12.2.5.20.2.3

Rewrite the expression.

$6 + 16 (\frac{1}{4} - \frac{0}{1})$

Step 12.2.5.20.2.4

Divide $0$ by $1$ .

$6 + 16 (\frac{1}{4} - 0)$

Step 12.2.5.21

Multiply $- 1$ by $0$ .

$6 + 16 (\frac{1}{4} + 0)$

Step 12.2.5.22

Add $\frac{1}{4}$ and $0$ .

$6 + 16 (\frac{1}{4})$

Step 12.2.5.23

Combine $16$ and $\frac{1}{4}$ .

$6 + \frac{16}{4}$

Step 12.2.5.24

Cancel the common factor of $16$ and $4$ .

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Step 12.2.5.24.1

Factor $4$ out of $16$ .

$6 + \frac{4 \cdot 4}{4}$

Step 12.2.5.24.2

Cancel the common factors.

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Step 12.2.5.24.2.1

Factor $4$ out of $4$ .

$6 + \frac{4 \cdot 4}{4 (1)}$

Step 12.2.5.24.2.2

Cancel the common factor.

$6 + \frac{4 \cdot 4}{4 \cdot 1}$

Step 12.2.5.24.2.3

Rewrite the expression.

$6 + \frac{4}{1}$

Step 12.2.5.24.2.4

Divide $4$ by $1$ .

$6 + 4$

Step 12.2.5.25

Add $6$ and $4$ .

$10$

Step 13