Calculus Examples

$\int_{0}^{3} 1 + 6 w^{2} - 10 w^{4} d w$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3} d w + \int_{0}^{3} 6 w^{2} d w + \int_{0}^{3} - 10 w^{4} d w$

Step 2

Apply the constant rule.

$w]_{0}^{3} + \int_{0}^{3} 6 w^{2} d w + \int_{0}^{3} - 10 w^{4} d w$

Step 3

Since $6$ is constant with respect to $w$ , move $6$ out of the integral.

$w]_{0}^{3} + 6 \int_{0}^{3} w^{2} d w + \int_{0}^{3} - 10 w^{4} d w$

Step 4

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

$w]_{0}^{3} + 6 (\frac{1}{3} w^{3}]_{0}^{3}) + \int_{0}^{3} - 10 w^{4} d w$

Step 5

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

$w]_{0}^{3} + 6 (\frac{w^{3}}{3}]_{0}^{3}) + \int_{0}^{3} - 10 w^{4} d w$

Step 6

Since $- 10$ is constant with respect to $w$ , move $- 10$ out of the integral.

$w]_{0}^{3} + 6 (\frac{w^{3}}{3}]_{0}^{3}) - 10 \int_{0}^{3} w^{4} d w$

Step 7

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

$w]_{0}^{3} + 6 (\frac{w^{3}}{3}]_{0}^{3}) - 10 (\frac{1}{5} w^{5}]_{0}^{3})$

Step 8

Simplify the answer.

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Step 8.1

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

$w]_{0}^{3} + 6 (\frac{w^{3}}{3}]_{0}^{3}) - 10 (\frac{w^{5}}{5}]_{0}^{3})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $w$ at $3$ and at $0$ .

$(3) + 0 + 6 (\frac{w^{3}}{3}]_{0}^{3}) - 10 (\frac{w^{5}}{5}]_{0}^{3})$

Step 8.2.2

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

$3 + 0 + 6 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 10 (\frac{w^{5}}{5}]_{0}^{3})$

Step 8.2.3

Evaluate $\frac{w^{5}}{5}$ at $3$ and at $0$ .

$3 + 0 + 6 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4

Simplify.

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Step 8.2.4.1

Add $3$ and $0$ .

$3 + 6 (\frac{3^{3}}{3} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.2

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

$3 + 6 (\frac{27}{3} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.3

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

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Step 8.2.4.3.1

Factor $3$ out of $27$ .

$3 + 6 (\frac{3 \cdot 9}{3} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.3.2

Cancel the common factors.

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Step 8.2.4.3.2.1

Factor $3$ out of $3$ .

$3 + 6 (\frac{3 \cdot 9}{3 (1)} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.3.2.2

Cancel the common factor.

$3 + 6 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.3.2.3

Rewrite the expression.

$3 + 6 (\frac{9}{1} - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.3.2.4

Divide $9$ by $1$ .

$3 + 6 (9 - \frac{0^{3}}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.4

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

$3 + 6 (9 - \frac{0}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.5

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

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Step 8.2.4.5.1

Factor $3$ out of $0$ .

$3 + 6 (9 - \frac{3 (0)}{3}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.5.2

Cancel the common factors.

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Step 8.2.4.5.2.1

Factor $3$ out of $3$ .

$3 + 6 (9 - \frac{3 \cdot 0}{3 \cdot 1}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.5.2.2

Cancel the common factor.

$3 + 6 (9 - \frac{3 \cdot 0}{3 \cdot 1}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.5.2.3

Rewrite the expression.

$3 + 6 (9 - \frac{0}{1}) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.5.2.4

Divide $0$ by $1$ .

$3 + 6 (9 - 0) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.6

Multiply $- 1$ by $0$ .

$3 + 6 (9 + 0) - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.7

Add $9$ and $0$ .

$3 + 6 \cdot 9 - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.8

Multiply $6$ by $9$ .

$3 + 54 - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.9

Add $3$ and $54$ .

$57 - 10 (\frac{3^{5}}{5} - \frac{0^{5}}{5})$

Step 8.2.4.10

Raise $3$ to the power of $5$ .

$57 - 10 (\frac{243}{5} - \frac{0^{5}}{5})$

Step 8.2.4.11

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

$57 - 10 (\frac{243}{5} - \frac{0}{5})$

Step 8.2.4.12

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

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Step 8.2.4.12.1

Factor $5$ out of $0$ .

$57 - 10 (\frac{243}{5} - \frac{5 (0)}{5})$

Step 8.2.4.12.2

Cancel the common factors.

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Step 8.2.4.12.2.1

Factor $5$ out of $5$ .

$57 - 10 (\frac{243}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 8.2.4.12.2.2

Cancel the common factor.

$57 - 10 (\frac{243}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 8.2.4.12.2.3

Rewrite the expression.

$57 - 10 (\frac{243}{5} - \frac{0}{1})$

Step 8.2.4.12.2.4

Divide $0$ by $1$ .

$57 - 10 (\frac{243}{5} - 0)$

Step 8.2.4.13

Multiply $- 1$ by $0$ .

$57 - 10 (\frac{243}{5} + 0)$

Step 8.2.4.14

Add $\frac{243}{5}$ and $0$ .

$57 - 10 (\frac{243}{5})$

Step 8.2.4.15

Combine $- 10$ and $\frac{243}{5}$ .

$57 + \frac{- 10 \cdot 243}{5}$

Step 8.2.4.16

Multiply $- 10$ by $243$ .

$57 + \frac{- 2430}{5}$

Step 8.2.4.17

Cancel the common factor of $- 2430$ and $5$ .

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Step 8.2.4.17.1

Factor $5$ out of $- 2430$ .

$57 + \frac{5 \cdot - 486}{5}$

Step 8.2.4.17.2

Cancel the common factors.

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Step 8.2.4.17.2.1

Factor $5$ out of $5$ .

$57 + \frac{5 \cdot - 486}{5 (1)}$

Step 8.2.4.17.2.2

Cancel the common factor.

$57 + \frac{5 \cdot - 486}{5 \cdot 1}$

Step 8.2.4.17.2.3

Rewrite the expression.

$57 + \frac{- 486}{1}$

Step 8.2.4.17.2.4

Divide $- 486$ by $1$ .

$57 - 486$

Step 8.2.4.18

Subtract $486$ from $57$ .

$- 429$

Step 9