Calculus Examples

$2 p k \int_{0}^{1} (5 x - x^{2}) x^{3} d x$

Step 1

Expand $(5 x - x^{2}) x^{3}$ .

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Step 1.1

Apply the distributive property.

$2 p k \int_{0}^{1} 5 x \cdot x^{3} - x^{2} x^{3} d x$

Step 1.2

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

$2 p k \int_{0}^{1} 5 (x^{1} x^{3}) - x^{2} x^{3} d x$

Step 1.3

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

$2 p k \int_{0}^{1} 5 x^{1 + 3} - x^{2} x^{3} d x$

Step 1.4

Add $1$ and $3$ .

$2 p k \int_{0}^{1} 5 x^{4} - x^{2} x^{3} d x$

Step 1.5

Factor out negative.

$2 p k \int_{0}^{1} 5 x^{4} - (x^{2} x^{3}) d x$

Step 1.6

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

$2 p k \int_{0}^{1} 5 x^{4} - x^{2 + 3} d x$

Step 1.7

Add $2$ and $3$ .

$2 p k \int_{0}^{1} 5 x^{4} - x^{5} d x$

Step 1.8

Reorder $5 x^{4}$ and $- x^{5}$ .

$2 p k \int_{0}^{1} - x^{5} + 5 x^{4} d x$

Step 2

Split the single integral into multiple integrals.

$2 p k (\int_{0}^{1} - x^{5} d x + \int_{0}^{1} 5 x^{4} d x)$

Step 3

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

$2 p k (- \int_{0}^{1} x^{5} d x + \int_{0}^{1} 5 x^{4} d x)$

Step 4

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$2 p k (- (\frac{1}{6} x^{6}]_{0}^{1}) + \int_{0}^{1} 5 x^{4} d x)$

Step 5

Combine $\frac{1}{6}$ and $x^{6}$ .

$2 p k (- (\frac{x^{6}}{6}]_{0}^{1}) + \int_{0}^{1} 5 x^{4} d x)$

Step 6

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

$2 p k (- (\frac{x^{6}}{6}]_{0}^{1}) + 5 \int_{0}^{1} x^{4} d x)$

Step 7

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$2 p k (- (\frac{x^{6}}{6}]_{0}^{1}) + 5 (\frac{1}{5} x^{5}]_{0}^{1}))$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$2 p k (- (\frac{x^{6}}{6}]_{0}^{1}) + 5 (\frac{x^{5}}{5}]_{0}^{1}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{6}}{6}$ at $1$ and at $0$ .

$2 p k (- ((\frac{1^{6}}{6}) - \frac{0^{6}}{6}) + 5 (\frac{x^{5}}{5}]_{0}^{1}))$

Step 8.2.2

Evaluate $\frac{x^{5}}{5}$ at $1$ and at $0$ .

$2 p k (- (\frac{1^{6}}{6} - \frac{0^{6}}{6}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3

Simplify.

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Step 8.2.3.1

One to any power is one.

$2 p k (- (\frac{1}{6} - \frac{0^{6}}{6}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.2

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

$2 p k (- (\frac{1}{6} - \frac{0}{6}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.3

Cancel the common factor of $0$ and $6$ .

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Step 8.2.3.3.1

Factor $6$ out of $0$ .

$2 p k (- (\frac{1}{6} - \frac{6 (0)}{6}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.3.2

Cancel the common factors.

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Step 8.2.3.3.2.1

Factor $6$ out of $6$ .

$2 p k (- (\frac{1}{6} - \frac{6 \cdot 0}{6 \cdot 1}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.3.2.2

Cancel the common factor.

$2 p k (- (\frac{1}{6} - \frac{6 \cdot 0}{6 \cdot 1}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.3.2.3

Rewrite the expression.

$2 p k (- (\frac{1}{6} - \frac{0}{1}) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.3.2.4

Divide $0$ by $1$ .

$2 p k (- (\frac{1}{6} - 0) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.4

Multiply $- 1$ by $0$ .

$2 p k (- (\frac{1}{6} + 0) + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.5

Add $\frac{1}{6}$ and $0$ .

$2 p k (- \frac{1}{6} + 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.6

One to any power is one.

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - \frac{0^{5}}{5}))$

Step 8.2.3.7

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

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - \frac{0}{5}))$

Step 8.2.3.8

Cancel the common factor of $0$ and $5$ .

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Step 8.2.3.8.1

Factor $5$ out of $0$ .

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - \frac{5 (0)}{5}))$

Step 8.2.3.8.2

Cancel the common factors.

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Step 8.2.3.8.2.1

Factor $5$ out of $5$ .

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - \frac{5 \cdot 0}{5 \cdot 1}))$

Step 8.2.3.8.2.2

Cancel the common factor.

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - \frac{5 \cdot 0}{5 \cdot 1}))$

Step 8.2.3.8.2.3

Rewrite the expression.

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - \frac{0}{1}))$

Step 8.2.3.8.2.4

Divide $0$ by $1$ .

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} - 0))$

Step 8.2.3.9

Multiply $- 1$ by $0$ .

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5} + 0))$

Step 8.2.3.10

Add $\frac{1}{5}$ and $0$ .

$2 p k (- \frac{1}{6} + 5 (\frac{1}{5}))$

Step 8.2.3.11

Combine $5$ and $\frac{1}{5}$ .

$2 p k (- \frac{1}{6} + \frac{5}{5})$

Step 8.2.3.12

Cancel the common factor of $5$ .

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Step 8.2.3.12.1

Cancel the common factor.

$2 p k (- \frac{1}{6} + \frac{5}{5})$

Step 8.2.3.12.2

Rewrite the expression.

$2 p k (- \frac{1}{6} + 1)$

Step 8.2.3.13

Write $1$ as a fraction with a common denominator.

$2 p k (- \frac{1}{6} + \frac{6}{6})$

Step 8.2.3.14

Combine the numerators over the common denominator.

$2 p k \frac{- 1 + 6}{6}$

Step 8.2.3.15

Add $- 1$ and $6$ .

$2 p k \frac{5}{6}$

Step 8.2.3.16

Combine $2$ and $\frac{5}{6}$ .

$p k \frac{2 \cdot 5}{6}$

Step 8.2.3.17

Multiply $2$ by $5$ .

$p k \frac{10}{6}$

Step 8.2.3.18

Combine $p$ and $\frac{10}{6}$ .

$k \frac{p \cdot 10}{6}$

Step 8.2.3.19

Combine $k$ and $\frac{p \cdot 10}{6}$ .

$\frac{k (p \cdot 10)}{6}$

Step 8.2.3.20

Move $10$ to the left of $p$ .

$\frac{k (10 \cdot p)}{6}$

Step 8.2.3.21

Move $10$ to the left of $k$ .

$\frac{10 \cdot k p}{6}$

Step 8.2.3.22

Cancel the common factor of $10$ and $6$ .

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Step 8.2.3.22.1

Factor $2$ out of $10 k p$ .

$\frac{2 (5 k p)}{6}$

Step 8.2.3.22.2

Cancel the common factors.

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Step 8.2.3.22.2.1

Factor $2$ out of $6$ .

$\frac{2 (5 k p)}{2 (3)}$

Step 8.2.3.22.2.2

Cancel the common factor.

$\frac{2 (5 k p)}{2 \cdot 3}$

Step 8.2.3.22.2.3

Rewrite the expression.

$\frac{5 k p}{3}$

$\frac{5}{3} k p$

Step 9

Simplify.

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Step 9.1

Combine $\frac{5}{3}$ and $k$ .

$\frac{5 k}{3} p$

Step 9.2

Combine $\frac{5 k}{3}$ and $p$ .

$\frac{5 k p}{3}$

Step 10