Calculus Examples

$\int_{0}^{3} (5 x^{2} - 4 x) d x$

Step 1

Remove parentheses.

$\int_{0}^{3} 5 x^{2} - 4 x d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{3} 5 x^{2} d x + \int_{0}^{3} - 4 x d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{3} x^{2} d x + \int_{0}^{3} - 4 x d x$

Step 4

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

$5 (\frac{1}{3} x^{3}]_{0}^{3}) + \int_{0}^{3} - 4 x d x$

Step 5

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

$5 (\frac{x^{3}}{3}]_{0}^{3}) + \int_{0}^{3} - 4 x d x$

Step 6

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

$5 (\frac{x^{3}}{3}]_{0}^{3}) - 4 \int_{0}^{3} x d x$

Step 7

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

$5 (\frac{x^{3}}{3}]_{0}^{3}) - 4 (\frac{1}{2} x^{2}]_{0}^{3})$

Step 8

Simplify the answer.

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Step 8.1

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

$5 (\frac{x^{3}}{3}]_{0}^{3}) - 4 (\frac{x^{2}}{2}]_{0}^{3})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

$5 ((\frac{3^{3}}{3}) - \frac{0^{3}}{3}) - 4 (\frac{x^{2}}{2}]_{0}^{3})$

Step 8.2.2

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

$5 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

$5 (\frac{27}{3} - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.2

Cancel the common factor of $27$ and $3$ .

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Step 8.2.3.2.1

Factor $3$ out of $27$ .

$5 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $3$ out of $3$ .

$5 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.2.2.2

Cancel the common factor.

$5 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.2.2.3

Rewrite the expression.

$5 (\frac{9}{1} - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.2.2.4

Divide $9$ by $1$ .

$5 (9 - \frac{0^{3}}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.3

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

$5 (9 - \frac{0}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.4

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

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Step 8.2.3.4.1

Factor $3$ out of $0$ .

$5 (9 - \frac{3 (0)}{3}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.4.2

Cancel the common factors.

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Step 8.2.3.4.2.1

Factor $3$ out of $3$ .

$5 (9 - \frac{3 \cdot 0}{3 \cdot 1}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.4.2.2

Cancel the common factor.

$5 (9 - \frac{3 \cdot 0}{3 \cdot 1}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.4.2.3

Rewrite the expression.

$5 (9 - \frac{0}{1}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.4.2.4

Divide $0$ by $1$ .

$5 (9 - 0) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.5

Multiply $- 1$ by $0$ .

$5 (9 + 0) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.6

Add $9$ and $0$ .

$5 \cdot 9 - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.7

Multiply $5$ by $9$ .

$45 - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 8.2.3.8

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

$45 - 4 (\frac{9}{2} - \frac{0^{2}}{2})$

Step 8.2.3.9

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

$45 - 4 (\frac{9}{2} - \frac{0}{2})$

Step 8.2.3.10

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

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Step 8.2.3.10.1

Factor $2$ out of $0$ .

$45 - 4 (\frac{9}{2} - \frac{2 (0)}{2})$

Step 8.2.3.10.2

Cancel the common factors.

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Step 8.2.3.10.2.1

Factor $2$ out of $2$ .

$45 - 4 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.10.2.2

Cancel the common factor.

$45 - 4 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.10.2.3

Rewrite the expression.

$45 - 4 (\frac{9}{2} - \frac{0}{1})$

Step 8.2.3.10.2.4

Divide $0$ by $1$ .

$45 - 4 (\frac{9}{2} - 0)$

Step 8.2.3.11

Multiply $- 1$ by $0$ .

$45 - 4 (\frac{9}{2} + 0)$

Step 8.2.3.12

Add $\frac{9}{2}$ and $0$ .

$45 - 4 (\frac{9}{2})$

Step 8.2.3.13

Combine $- 4$ and $\frac{9}{2}$ .

$45 + \frac{- 4 \cdot 9}{2}$

Step 8.2.3.14

Multiply $- 4$ by $9$ .

$45 + \frac{- 36}{2}$

Step 8.2.3.15

Cancel the common factor of $- 36$ and $2$ .

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Step 8.2.3.15.1

Factor $2$ out of $- 36$ .

$45 + \frac{2 \cdot - 18}{2}$

Step 8.2.3.15.2

Cancel the common factors.

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Step 8.2.3.15.2.1

Factor $2$ out of $2$ .

$45 + \frac{2 \cdot - 18}{2 (1)}$

Step 8.2.3.15.2.2

Cancel the common factor.

$45 + \frac{2 \cdot - 18}{2 \cdot 1}$

Step 8.2.3.15.2.3

Rewrite the expression.

$45 + \frac{- 18}{1}$

Step 8.2.3.15.2.4

Divide $- 18$ by $1$ .

$45 - 18$

Step 8.2.3.16

Subtract $18$ from $45$ .

$27$

Step 9