Calculus Examples

$\int 5 m (12 m^{3} - 10 m) d m$

Step 1

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

$5 \int m (12 m^{3} - 10 m) d m$

Step 2

Simplify.

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Step 2.1

Apply the distributive property.

$5 \int m (12 m^{3}) + m (- 10 m) d m$

Step 2.2

Reorder $m$ and $12$ .

$5 \int 12 \cdot m \cdot m^{3} + m (- 10 m) d m$

Step 2.3

Reorder $m$ and $- 10$ .

$5 \int 12 \cdot m \cdot m^{3} - 10 \cdot m \cdot m d m$

Step 2.4

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

$5 \int 12 (m^{1} m^{3}) - 10 m \cdot m d m$

Step 2.5

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

$5 \int 12 m^{1 + 3} - 10 m \cdot m d m$

Step 2.6

Add $1$ and $3$ .

$5 \int 12 m^{4} - 10 m \cdot m d m$

Step 2.7

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

$5 \int 12 m^{4} - 10 (m^{1} m) d m$

Step 2.8

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

$5 \int 12 m^{4} - 10 (m^{1} m^{1}) d m$

Step 2.9

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

$5 \int 12 m^{4} - 10 m^{1 + 1} d m$

Step 2.10

Add $1$ and $1$ .

$5 \int 12 m^{4} - 10 m^{2} d m$

Step 3

Split the single integral into multiple integrals.

$5 (\int 12 m^{4} d m + \int - 10 m^{2} d m)$

Step 4

Since $12$ is constant with respect to $m$ , move $12$ out of the integral.

$5 (12 \int m^{4} d m + \int - 10 m^{2} d m)$

Step 5

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

$5 (12 (\frac{1}{5} m^{5} + C) + \int - 10 m^{2} d m)$

Step 6

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

$5 (12 (\frac{1}{5} m^{5} + C) - 10 \int m^{2} d m)$

Step 7

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

$5 (12 (\frac{1}{5} m^{5} + C) - 10 (\frac{1}{3} m^{3} + C))$

Step 8

Simplify.

$5 (\frac{12 m^{5}}{5} - \frac{10 m^{3}}{3}) + C$

Step 9

Reorder terms.

$5 (\frac{12}{5} m^{5} - \frac{10}{3} m^{3}) + C$