Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

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

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

Step 11.2.2

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $0$ .

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

Step 11.2.3

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

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

Step 11.2.4

Simplify.

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Step 11.2.4.1

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

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

Step 11.2.4.2

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

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Step 11.2.4.2.1

Factor $4$ out of $16$ .

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

Step 11.2.4.2.2

Cancel the common factors.

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Step 11.2.4.2.2.1

Factor $4$ out of $4$ .

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

Step 11.2.4.2.2.2

Cancel the common factor.

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

Step 11.2.4.2.2.3

Rewrite the expression.

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

Step 11.2.4.2.2.4

Divide $4$ by $1$ .

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

Step 11.2.4.3

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

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

Step 11.2.4.4

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

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Step 11.2.4.4.1

Factor $4$ out of $0$ .

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

Step 11.2.4.4.2

Cancel the common factors.

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Step 11.2.4.4.2.1

Factor $4$ out of $4$ .

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

Step 11.2.4.4.2.2

Cancel the common factor.

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

Step 11.2.4.4.2.3

Rewrite the expression.

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

Step 11.2.4.4.2.4

Divide $0$ by $1$ .

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

Step 11.2.4.5

Multiply $- 1$ by $0$ .

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

Step 11.2.4.6

Add $4$ and $0$ .

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

Step 11.2.4.7

Multiply $4$ by $4$ .

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

Step 11.2.4.8

Raise $2$ to the power of $3$ .

$16 - 3 (\frac{8}{3} - \frac{0^{3}}{3}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.9

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

$16 - 3 (\frac{8}{3} - \frac{0}{3}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.10

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

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Step 11.2.4.10.1

Factor $3$ out of $0$ .

$16 - 3 (\frac{8}{3} - \frac{3 (0)}{3}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.10.2

Cancel the common factors.

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Step 11.2.4.10.2.1

Factor $3$ out of $3$ .

$16 - 3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.10.2.2

Cancel the common factor.

$16 - 3 (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.10.2.3

Rewrite the expression.

$16 - 3 (\frac{8}{3} - \frac{0}{1}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.10.2.4

Divide $0$ by $1$ .

$16 - 3 (\frac{8}{3} - 0) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.11

Multiply $- 1$ by $0$ .

$16 - 3 (\frac{8}{3} + 0) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.12

Add $\frac{8}{3}$ and $0$ .

$16 - 3 (\frac{8}{3}) + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.13

Combine $- 3$ and $\frac{8}{3}$ .

$16 + \frac{- 3 \cdot 8}{3} + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.14

Multiply $- 3$ by $8$ .

$16 + \frac{- 24}{3} + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.15

Cancel the common factor of $- 24$ and $3$ .

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Step 11.2.4.15.1

Factor $3$ out of $- 24$ .

$16 + \frac{3 \cdot - 8}{3} + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.15.2

Cancel the common factors.

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Step 11.2.4.15.2.1

Factor $3$ out of $3$ .

$16 + \frac{3 \cdot - 8}{3 (1)} + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.15.2.2

Cancel the common factor.

$16 + \frac{3 \cdot - 8}{3 \cdot 1} + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.15.2.3

Rewrite the expression.

$16 + \frac{- 8}{1} + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.15.2.4

Divide $- 8$ by $1$ .

$16 - 8 + 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 11.2.4.16

Subtract $8$ from $16$ .

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

Step 11.2.4.17

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

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

Step 11.2.4.18

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

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Step 11.2.4.18.1

Factor $2$ out of $4$ .

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

Step 11.2.4.18.2

Cancel the common factors.

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Step 11.2.4.18.2.1

Factor $2$ out of $2$ .

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

Step 11.2.4.18.2.2

Cancel the common factor.

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

Step 11.2.4.18.2.3

Rewrite the expression.

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

Step 11.2.4.18.2.4

Divide $2$ by $1$ .

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

Step 11.2.4.19

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

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

Step 11.2.4.20

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

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Step 11.2.4.20.1

Factor $2$ out of $0$ .

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

Step 11.2.4.20.2

Cancel the common factors.

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Step 11.2.4.20.2.1

Factor $2$ out of $2$ .

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

Step 11.2.4.20.2.2

Cancel the common factor.

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

Step 11.2.4.20.2.3

Rewrite the expression.

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

Step 11.2.4.20.2.4

Divide $0$ by $1$ .

$8 + 2 (2 - 0)$

Step 11.2.4.21

Multiply $- 1$ by $0$ .

$8 + 2 (2 + 0)$

Step 11.2.4.22

Add $2$ and $0$ .

$8 + 2 \cdot 2$

Step 11.2.4.23

Multiply $2$ by $2$ .

$8 + 4$

Step 11.2.4.24

Add $8$ and $4$ .

$12$

Step 12