Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

$4 (\frac{x^{6}}{6}]_{0}^{1}) - 5 (\frac{1}{5} x^{5}]_{0}^{1})$

Step 7

Simplify the answer.

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Step 7.1

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

$4 (\frac{x^{6}}{6}]_{0}^{1}) - 5 (\frac{x^{5}}{5}]_{0}^{1})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

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

$4 ((\frac{1^{6}}{6}) - \frac{0^{6}}{6}) - 5 (\frac{x^{5}}{5}]_{0}^{1})$

Step 7.2.2

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

$4 (\frac{1^{6}}{6} - \frac{0^{6}}{6}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

One to any power is one.

$4 (\frac{1}{6} - \frac{0^{6}}{6}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.2

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

$4 (\frac{1}{6} - \frac{0}{6}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.3

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

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Step 7.2.3.3.1

Factor $6$ out of $0$ .

$4 (\frac{1}{6} - \frac{6 (0)}{6}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.3.2

Cancel the common factors.

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Step 7.2.3.3.2.1

Factor $6$ out of $6$ .

$4 (\frac{1}{6} - \frac{6 \cdot 0}{6 \cdot 1}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.3.2.2

Cancel the common factor.

$4 (\frac{1}{6} - \frac{6 \cdot 0}{6 \cdot 1}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.3.2.3

Rewrite the expression.

$4 (\frac{1}{6} - \frac{0}{1}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.3.2.4

Divide $0$ by $1$ .

$4 (\frac{1}{6} - 0) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.4

Multiply $- 1$ by $0$ .

$4 (\frac{1}{6} + 0) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.5

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

$4 (\frac{1}{6}) - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.6

Combine $4$ and $\frac{1}{6}$ .

$\frac{4}{6} - 5 (\frac{1^{5}}{5} - \frac{0^{5}}{5})$

Step 7.2.3.7

Cancel the common factor of $4$ and $6$ .

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Step 7.2.3.7.1

Factor $2$ out of $4$ .

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

Step 7.2.3.7.2

Cancel the common factors.

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Step 7.2.3.7.2.1

Factor $2$ out of $6$ .

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

Step 7.2.3.7.2.2

Cancel the common factor.

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

Step 7.2.3.7.2.3

Rewrite the expression.

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

Step 7.2.3.8

One to any power is one.

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

Step 7.2.3.9

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

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

Step 7.2.3.10

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

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Step 7.2.3.10.1

Factor $5$ out of $0$ .

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

Step 7.2.3.10.2

Cancel the common factors.

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Step 7.2.3.10.2.1

Factor $5$ out of $5$ .

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

Step 7.2.3.10.2.2

Cancel the common factor.

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

Step 7.2.3.10.2.3

Rewrite the expression.

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

Step 7.2.3.10.2.4

Divide $0$ by $1$ .

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

Step 7.2.3.11

Multiply $- 1$ by $0$ .

$\frac{2}{3} - 5 (\frac{1}{5} + 0)$

Step 7.2.3.12

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

$\frac{2}{3} - 5 (\frac{1}{5})$

Step 7.2.3.13

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

$\frac{2}{3} + \frac{- 5}{5}$

Step 7.2.3.14

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

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Step 7.2.3.14.1

Factor $5$ out of $- 5$ .

$\frac{2}{3} + \frac{5 \cdot - 1}{5}$

Step 7.2.3.14.2

Cancel the common factors.

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Step 7.2.3.14.2.1

Factor $5$ out of $5$ .

$\frac{2}{3} + \frac{5 \cdot - 1}{5 (1)}$

Step 7.2.3.14.2.2

Cancel the common factor.

$\frac{2}{3} + \frac{5 \cdot - 1}{5 \cdot 1}$

Step 7.2.3.14.2.3

Rewrite the expression.

$\frac{2}{3} + \frac{- 1}{1}$

Step 7.2.3.14.2.4

Divide $- 1$ by $1$ .

$\frac{2}{3} - 1$

Step 7.2.3.15

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} - 1 \cdot \frac{3}{3}$

Step 7.2.3.16

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

$\frac{2}{3} + \frac{- 1 \cdot 3}{3}$

Step 7.2.3.17

Combine the numerators over the common denominator.

$\frac{2 - 1 \cdot 3}{3}$

Step 7.2.3.18

Simplify the numerator.

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Step 7.2.3.18.1

Multiply $- 1$ by $3$ .

$\frac{2 - 3}{3}$

Step 7.2.3.18.2

Subtract $3$ from $2$ .

$\frac{- 1}{3}$

Step 7.2.3.19

Move the negative in front of the fraction.

$- \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{3}$

Decimal Form:

$- 0.\overline{3}$

Step 9