Calculus Examples

$(x^{3} + x^{2}) v (\frac{5 x^{4}}{2} + \frac{10 x^{3}}{3})$

Step 1

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

$v \int (x^{3} + x^{2}) (\frac{5 x^{4}}{2} + \frac{10 x^{3}}{3}) d x$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$v \int x^{3} (\frac{5 x^{4}}{2} + \frac{10 x^{3}}{3}) + x^{2} (\frac{5 x^{4}}{2} + \frac{10 x^{3}}{3}) d x$

Step 2.2

Apply the distributive property.

$v \int x^{3} \frac{5 x^{4}}{2} + x^{3} \frac{10 x^{3}}{3} + x^{2} (\frac{5 x^{4}}{2} + \frac{10 x^{3}}{3}) d x$

Step 2.3

Apply the distributive property.

$v \int x^{3} \frac{5 x^{4}}{2} + x^{3} \frac{10 x^{3}}{3} + x^{2} \frac{5 x^{4}}{2} + x^{2} \frac{10 x^{3}}{3} d x$

Step 2.4

Combine $x^{3}$ and $\frac{5 x^{4}}{2}$ .

$v \int \frac{x^{3} (5 x^{4})}{2} + x^{3} \frac{10 x^{3}}{3} + x^{2} \frac{5 x^{4}}{2} + x^{2} \frac{10 x^{3}}{3} d x$

Step 2.5

Combine $x^{3}$ and $\frac{10 x^{3}}{3}$ .

$v \int \frac{x^{3} (5 x^{4})}{2} + \frac{x^{3} (10 x^{3})}{3} + x^{2} \frac{5 x^{4}}{2} + x^{2} \frac{10 x^{3}}{3} d x$

Step 2.6

Combine $x^{2}$ and $\frac{5 x^{4}}{2}$ .

$v \int \frac{x^{3} (5 x^{4})}{2} + \frac{x^{3} (10 x^{3})}{3} + \frac{x^{2} (5 x^{4})}{2} + x^{2} \frac{10 x^{3}}{3} d x$

Step 2.7

Combine $x^{2}$ and $\frac{10 x^{3}}{3}$ .

$v \int \frac{x^{3} (5 x^{4})}{2} + \frac{x^{3} (10 x^{3})}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3

Simplify.

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Step 3.1

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 3.1.1

Move $x^{4}$ .

$v \int \frac{x^{4} x^{3} \cdot 5}{2} + \frac{x^{3} (10 x^{3})}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.1.2

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

$v \int \frac{x^{4 + 3} \cdot 5}{2} + \frac{x^{3} (10 x^{3})}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.1.3

Add $4$ and $3$ .

$v \int \frac{x^{7} \cdot 5}{2} + \frac{x^{3} (10 x^{3})}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.2

Move $5$ to the left of $x^{7}$ .

$v \int \frac{5 \cdot x^{7}}{2} + \frac{x^{3} (10 x^{3})}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.3

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 3.3.1

Move $x^{3}$ .

$v \int \frac{5 x^{7}}{2} + \frac{x^{3} x^{3} \cdot 10}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.3.2

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

$v \int \frac{5 x^{7}}{2} + \frac{x^{3 + 3} \cdot 10}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.3.3

Add $3$ and $3$ .

$v \int \frac{5 x^{7}}{2} + \frac{x^{6} \cdot 10}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.4

Move $10$ to the left of $x^{6}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 \cdot x^{6}}{3} + \frac{x^{2} (5 x^{4})}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.5

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 3.5.1

Move $x^{4}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6}}{3} + \frac{x^{4} x^{2} \cdot 5}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.5.2

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

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6}}{3} + \frac{x^{4 + 2} \cdot 5}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.5.3

Add $4$ and $2$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6}}{3} + \frac{x^{6} \cdot 5}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.6

Move $5$ to the left of $x^{6}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6}}{3} + \frac{5 \cdot x^{6}}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.7

To write $\frac{10 x^{6}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6}}{3} \cdot \frac{2}{2} + \frac{5 x^{6}}{2} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.8

To write $\frac{5 x^{6}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6}}{3} \cdot \frac{2}{2} + \frac{5 x^{6}}{2} \cdot \frac{3}{3} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.9

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.1

Multiply $\frac{10 x^{6}}{3}$ by $\frac{2}{2}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6} \cdot 2}{3 \cdot 2} + \frac{5 x^{6}}{2} \cdot \frac{3}{3} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.9.2

Multiply $3$ by $2$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6} \cdot 2}{6} + \frac{5 x^{6}}{2} \cdot \frac{3}{3} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.9.3

Multiply $\frac{5 x^{6}}{2}$ by $\frac{3}{3}$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6} \cdot 2}{6} + \frac{5 x^{6} \cdot 3}{2 \cdot 3} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.9.4

Multiply $2$ by $3$ .

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6} \cdot 2}{6} + \frac{5 x^{6} \cdot 3}{6} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.10

Combine the numerators over the common denominator.

$v \int \frac{5 x^{7}}{2} + \frac{10 x^{6} \cdot 2 + 5 x^{6} \cdot 3}{6} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.11

Multiply $2$ by $10$ .

$v \int \frac{5 x^{7}}{2} + \frac{20 x^{6} + 5 x^{6} \cdot 3}{6} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.12

Multiply $3$ by $5$ .

$v \int \frac{5 x^{7}}{2} + \frac{20 x^{6} + 15 x^{6}}{6} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.13

Add $20 x^{6}$ and $15 x^{6}$ .

$v \int \frac{5 x^{7}}{2} + \frac{35 x^{6}}{6} + \frac{x^{2} (10 x^{3})}{3} d x$

Step 3.14

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.14.1

Move $x^{3}$ .

$v \int \frac{5 x^{7}}{2} + \frac{35 x^{6}}{6} + \frac{x^{3} x^{2} \cdot 10}{3} d x$

Step 3.14.2

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

$v \int \frac{5 x^{7}}{2} + \frac{35 x^{6}}{6} + \frac{x^{3 + 2} \cdot 10}{3} d x$

Step 3.14.3

Add $3$ and $2$ .

$v \int \frac{5 x^{7}}{2} + \frac{35 x^{6}}{6} + \frac{x^{5} \cdot 10}{3} d x$

Step 3.15

Move $10$ to the left of $x^{5}$ .

$v \int \frac{5 x^{7}}{2} + \frac{35 x^{6}}{6} + \frac{10 x^{5}}{3} d x$

Step 4

Split the single integral into multiple integrals.

$v (\int \frac{5 x^{7}}{2} d x + \int \frac{35 x^{6}}{6} d x + \int \frac{10 x^{5}}{3} d x)$

Step 5

Since $\frac{5}{2}$ is constant with respect to $x$ , move $\frac{5}{2}$ out of the integral.

$v (\frac{5}{2} \int x^{7} d x + \int \frac{35 x^{6}}{6} d x + \int \frac{10 x^{5}}{3} d x)$

Step 6

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$v (\frac{5}{2} (\frac{1}{8} x^{8} + C) + \int \frac{35 x^{6}}{6} d x + \int \frac{10 x^{5}}{3} d x)$

Step 7

Combine $\frac{1}{8}$ and $x^{8}$ .

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \int \frac{35 x^{6}}{6} d x + \int \frac{10 x^{5}}{3} d x)$

Step 8

Since $\frac{35}{6}$ is constant with respect to $x$ , move $\frac{35}{6}$ out of the integral.

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \frac{35}{6} \int x^{6} d x + \int \frac{10 x^{5}}{3} d x)$

Step 9

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \frac{35}{6} (\frac{1}{7} x^{7} + C) + \int \frac{10 x^{5}}{3} d x)$

Step 10

Combine $\frac{1}{7}$ and $x^{7}$ .

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \frac{35}{6} (\frac{x^{7}}{7} + C) + \int \frac{10 x^{5}}{3} d x)$

Step 11

Since $\frac{10}{3}$ is constant with respect to $x$ , move $\frac{10}{3}$ out of the integral.

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \frac{35}{6} (\frac{x^{7}}{7} + C) + \frac{10}{3} \int x^{5} d x)$

Step 12

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

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \frac{35}{6} (\frac{x^{7}}{7} + C) + \frac{10}{3} (\frac{1}{6} x^{6} + C))$

Step 13

Simplify.

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Step 13.1

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

$v (\frac{5}{2} (\frac{x^{8}}{8} + C) + \frac{35}{6} (\frac{x^{7}}{7} + C) + \frac{10}{3} (\frac{x^{6}}{6} + C))$

Step 13.2

Simplify.

$v (\frac{5 x^{8}}{16} + \frac{5 x^{7}}{6} + \frac{5 x^{6}}{9}) + C$

Step 13.3

Reorder terms.

$v (\frac{5}{16} x^{8} + \frac{5}{6} x^{7} + \frac{5}{9} x^{6}) + C$