Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.1

Move $x^{2}$ .

$\int_{0}^{3} \frac{1}{4} (x^{2} x^{2}) \cdot 2 (3 - x) d x$

Step 1.1.2

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

$\int_{0}^{3} \frac{1}{4} x^{2 + 2} \cdot 2 (3 - x) d x$

Step 1.1.3

Add $2$ and $2$ .

$\int_{0}^{3} \frac{1}{4} x^{4} \cdot 2 (3 - x) d x$

Step 1.2

Combine $\frac{1}{4}$ and $x^{4}$ .

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

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Factor $2$ out of $4$ .

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

Step 1.3.2

Cancel the common factor.

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

Step 1.3.3

Rewrite the expression.

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

Step 1.4

Apply the distributive property.

$\int_{0}^{3} \frac{x^{4}}{2} \cdot 3 + \frac{x^{4}}{2} (- x) d x$

Step 1.5

Combine $\frac{x^{4}}{2}$ and $3$ .

$\int_{0}^{3} \frac{x^{4} \cdot 3}{2} + \frac{x^{4}}{2} (- x) d x$

Step 1.6

Rewrite using the commutative property of multiplication.

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

Step 1.7

Simplify each term.

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Step 1.7.1

Move $3$ to the left of $x^{4}$ .

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

Step 1.7.2

Multiply $- \frac{x^{4}}{2} x$ .

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Step 1.7.2.1

Combine $x$ and $\frac{x^{4}}{2}$ .

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

Step 1.7.2.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 1.7.2.2.1

Multiply $x$ by $x^{4}$ .

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Step 1.7.2.2.1.1

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

$\int_{0}^{3} \frac{3 x^{4}}{2} - \frac{x^{1} x^{4}}{2} d x$

Step 1.7.2.2.1.2

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

$\int_{0}^{3} \frac{3 x^{4}}{2} - \frac{x^{1 + 4}}{2} d x$

Step 1.7.2.2.2

Add $1$ and $4$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

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

Step 4

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

$\frac{3}{2} \frac{1}{5} x^{5}]_{0}^{3} + \int_{0}^{3} - \frac{x^{5}}{2} d x$

Step 5

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

$\frac{3}{2} \frac{1}{5} x^{5}]_{0}^{3} - \int_{0}^{3} \frac{x^{5}}{2} d x$

Step 6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 7

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

$\frac{3}{2} \frac{1}{5} x^{5}]_{0}^{3} - \frac{1}{2} \frac{1}{6} x^{6}]_{0}^{3}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{5} x^{5}$ at $3$ and at $0$ .

$\frac{3}{2} ((\frac{1}{5} \cdot 3^{5}) - \frac{1}{5} \cdot 0^{5}) - \frac{1}{2} \frac{1}{6} x^{6}]_{0}^{3}$

Step 8.2

Evaluate $\frac{1}{6} x^{6}$ at $3$ and at $0$ .

$\frac{3}{2} (\frac{1}{5} \cdot 3^{5} - \frac{1}{5} \cdot 0^{5}) - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3

Simplify.

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Step 8.3.1

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

$\frac{3}{2} (\frac{1}{5} \cdot 243 - \frac{1}{5} \cdot 0^{5}) - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.2

Combine $\frac{1}{5}$ and $243$ .

$\frac{3}{2} (\frac{243}{5} - \frac{1}{5} \cdot 0^{5}) - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.3

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

$\frac{3}{2} (\frac{243}{5} - \frac{1}{5} \cdot 0) - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.4

Multiply $0$ by $- 1$ .

$\frac{3}{2} (\frac{243}{5} + 0 (\frac{1}{5})) - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.5

Multiply $0$ by $\frac{1}{5}$ .

$\frac{3}{2} (\frac{243}{5} + 0) - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.6

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

$\frac{3}{2} \cdot \frac{243}{5} - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.7

Multiply $\frac{3}{2}$ by $\frac{243}{5}$ .

$\frac{3 \cdot 243}{2 \cdot 5} - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.8

Multiply $3$ by $243$ .

$\frac{729}{2 \cdot 5} - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.9

Multiply $2$ by $5$ .

$\frac{729}{10} - \frac{1}{2} (\frac{1}{6} \cdot 3^{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.10

Raise $3$ to the power of $6$ .

$\frac{729}{10} - \frac{1}{2} (\frac{1}{6} \cdot 729 - \frac{1}{6} \cdot 0^{6})$

Step 8.3.11

Combine $\frac{1}{6}$ and $729$ .

$\frac{729}{10} - \frac{1}{2} (\frac{729}{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.12

Cancel the common factor of $729$ and $6$ .

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Step 8.3.12.1

Factor $3$ out of $729$ .

$\frac{729}{10} - \frac{1}{2} (\frac{3 (243)}{6} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.12.2

Cancel the common factors.

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Step 8.3.12.2.1

Factor $3$ out of $6$ .

$\frac{729}{10} - \frac{1}{2} (\frac{3 \cdot 243}{3 \cdot 2} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.12.2.2

Cancel the common factor.

$\frac{729}{10} - \frac{1}{2} (\frac{3 \cdot 243}{3 \cdot 2} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.12.2.3

Rewrite the expression.

$\frac{729}{10} - \frac{1}{2} (\frac{243}{2} - \frac{1}{6} \cdot 0^{6})$

Step 8.3.13

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

$\frac{729}{10} - \frac{1}{2} (\frac{243}{2} - \frac{1}{6} \cdot 0)$

Step 8.3.14

Multiply $0$ by $- 1$ .

$\frac{729}{10} - \frac{1}{2} (\frac{243}{2} + 0 (\frac{1}{6}))$

Step 8.3.15

Multiply $0$ by $\frac{1}{6}$ .

$\frac{729}{10} - \frac{1}{2} (\frac{243}{2} + 0)$

Step 8.3.16

Add $\frac{243}{2}$ and $0$ .

$\frac{729}{10} - \frac{1}{2} \cdot \frac{243}{2}$

Step 8.3.17

Multiply $\frac{243}{2}$ by $\frac{1}{2}$ .

$\frac{729}{10} - \frac{243}{2 \cdot 2}$

Step 8.3.18

Multiply $2$ by $2$ .

$\frac{729}{10} - \frac{243}{4}$

Step 8.3.19

To write $\frac{729}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{729}{10} \cdot \frac{2}{2} - \frac{243}{4}$

Step 8.3.20

To write $- \frac{243}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{729}{10} \cdot \frac{2}{2} - \frac{243}{4} \cdot \frac{5}{5}$

Step 8.3.21

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.3.21.1

Multiply $\frac{729}{10}$ by $\frac{2}{2}$ .

$\frac{729 \cdot 2}{10 \cdot 2} - \frac{243}{4} \cdot \frac{5}{5}$

Step 8.3.21.2

Multiply $10$ by $2$ .

$\frac{729 \cdot 2}{20} - \frac{243}{4} \cdot \frac{5}{5}$

Step 8.3.21.3

Multiply $\frac{243}{4}$ by $\frac{5}{5}$ .

$\frac{729 \cdot 2}{20} - \frac{243 \cdot 5}{4 \cdot 5}$

Step 8.3.21.4

Multiply $4$ by $5$ .

$\frac{729 \cdot 2}{20} - \frac{243 \cdot 5}{20}$

Step 8.3.22

Combine the numerators over the common denominator.

$\frac{729 \cdot 2 - 243 \cdot 5}{20}$

Step 8.3.23

Simplify the numerator.

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Step 8.3.23.1

Multiply $729$ by $2$ .

$\frac{1458 - 243 \cdot 5}{20}$

Step 8.3.23.2

Multiply $- 243$ by $5$ .

$\frac{1458 - 1215}{20}$

Step 8.3.23.3

Subtract $1215$ from $1458$ .

$\frac{243}{20}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{243}{20}$

Decimal Form:

$12.15$

Mixed Number Form:

$12 \frac{3}{20}$

Step 10