Calculus Examples

$\int 15 s^{4} - 8 s d s$

Step 1

Split the single integral into multiple integrals.

$\int 15 s^{4} d s + \int - 8 s d s$

Step 2

Since $15$ is constant with respect to $s$ , move $15$ out of the integral.

$15 \int s^{4} d s + \int - 8 s d s$

Step 3

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

$15 (\frac{1}{5} s^{5} + C) + \int - 8 s d s$

Step 4

Since $- 8$ is constant with respect to $s$ , move $- 8$ out of the integral.

$15 (\frac{1}{5} s^{5} + C) - 8 \int s d s$

Step 5

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

$15 (\frac{1}{5} s^{5} + C) - 8 (\frac{1}{2} s^{2} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

$15 (\frac{1}{5}) s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $15$ and $\frac{1}{5}$ .

$\frac{15}{5} s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2.2

Cancel the common factor of $15$ and $5$ .

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Step 6.2.2.1

Factor $5$ out of $15$ .

$\frac{5 \cdot 3}{5} s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 3}{5 (1)} s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2.2.2.2

Cancel the common factor.

$\frac{5 \cdot 3}{5 \cdot 1} s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2.2.2.3

Rewrite the expression.

$\frac{3}{1} s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2.2.2.4

Divide $3$ by $1$ .

$3 s^{5} - 8 (\frac{1}{2}) s^{2} + C$

Step 6.2.3

Combine $- 8$ and $\frac{1}{2}$ .

$3 s^{5} + \frac{- 8}{2} s^{2} + C$

Step 6.2.4

Cancel the common factor of $- 8$ and $2$ .

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Step 6.2.4.1

Factor $2$ out of $- 8$ .

$3 s^{5} + \frac{2 \cdot - 4}{2} s^{2} + C$

Step 6.2.4.2

Cancel the common factors.

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Step 6.2.4.2.1

Factor $2$ out of $2$ .

$3 s^{5} + \frac{2 \cdot - 4}{2 (1)} s^{2} + C$

Step 6.2.4.2.2

Cancel the common factor.

$3 s^{5} + \frac{2 \cdot - 4}{2 \cdot 1} s^{2} + C$

Step 6.2.4.2.3

Rewrite the expression.

$3 s^{5} + \frac{- 4}{1} s^{2} + C$

Step 6.2.4.2.4

Divide $- 4$ by $1$ .

$3 s^{5} - 4 s^{2} + C$