Calculus Examples

$\int_{0}^{k} (2 k x - x^{2}) d x = 18$

Step 1

Remove parentheses.

$\int_{0}^{k} 2 k x - x^{2} d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

$2 k \frac{1}{2} x^{2}]_{0}^{k} + \int_{0}^{k} - x^{2} d x$

Step 5

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

$2 k \frac{x^{2}}{2}]_{0}^{k} + \int_{0}^{k} - x^{2} d x$

Step 6

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

$2 k \frac{x^{2}}{2}]_{0}^{k} - \int_{0}^{k} x^{2} d x$

Step 7

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

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

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

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

Step 8.2.3.2

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

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Step 8.2.3.2.1

Factor $2$ out of $0$ .

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

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.3.2.2.2

Cancel the common factor.

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

Step 8.2.3.2.2.3

Rewrite the expression.

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

Step 8.2.3.2.2.4

Divide $0$ by $1$ .

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

Step 8.2.3.3

Multiply $- 1$ by $0$ .

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

Step 8.2.3.4

Add $\frac{k^{2}}{2}$ and $0$ .

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

Step 8.2.3.5

Combine $2$ and $\frac{k^{2}}{2}$ .

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

Step 8.2.3.6

Combine $k$ and $\frac{2 k^{2}}{2}$ .

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

Step 8.2.3.7

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

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

Step 8.2.3.8

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

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

Step 8.2.3.9

Add $1$ and $2$ .

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

Step 8.2.3.10

Cancel the common factor of $2$ .

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Step 8.2.3.10.1

Cancel the common factor.

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

Step 8.2.3.10.2

Divide $k^{3}$ by $1$ .

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

Step 8.2.3.11

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

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

Step 8.2.3.12

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

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Step 8.2.3.12.1

Factor $3$ out of $0$ .

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

Step 8.2.3.12.2

Cancel the common factors.

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Step 8.2.3.12.2.1

Factor $3$ out of $3$ .

$k^{3} - (\frac{k^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.3.12.2.2

Cancel the common factor.

$k^{3} - (\frac{k^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 8.2.3.12.2.3

Rewrite the expression.

$k^{3} - (\frac{k^{3}}{3} - \frac{0}{1})$

Step 8.2.3.12.2.4

Divide $0$ by $1$ .

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

Step 8.2.3.13

Multiply $- 1$ by $0$ .

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

Step 8.2.3.14

Add $\frac{k^{3}}{3}$ and $0$ .

$k^{3} - \frac{k^{3}}{3}$

Step 8.2.3.15

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

$k^{3} \cdot \frac{3}{3} - \frac{k^{3}}{3}$

Step 8.2.3.16

Combine $k^{3}$ and $\frac{3}{3}$ .

$\frac{k^{3} \cdot 3}{3} - \frac{k^{3}}{3}$

Step 8.2.3.17

Combine the numerators over the common denominator.

$\frac{k^{3} \cdot 3 - k^{3}}{3}$

Step 8.2.3.18

Move $3$ to the left of $k^{3}$ .

$\frac{3 \cdot k^{3} - k^{3}}{3}$

Step 8.2.3.19

Subtract $k^{3}$ from $3 k^{3}$ .

$\frac{2 k^{3}}{3}$

Step 8.3

Reorder terms.

$\frac{2}{3} k^{3}$

Step 9

Combine $\frac{2}{3}$ and $k^{3}$ .

$\frac{2 k^{3}}{3}$