Calculus Examples

$3 - 12 t^{3} - 9 t^{5}$

Step 1

Split the single integral into multiple integrals.

$\int 3 d t + \int - 12 t^{3} d t + \int - 9 t^{5} d t$

Step 2

Apply the constant rule.

$3 t + C + \int - 12 t^{3} d t + \int - 9 t^{5} d t$

Step 3

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

$3 t + C - 12 \int t^{3} d t + \int - 9 t^{5} d t$

Step 4

By the Power Rule, the integral of $t^{3}$ with respect to $t$ is $\frac{1}{4} t^{4}$ .

$3 t + C - 12 (\frac{1}{4} t^{4} + C) + \int - 9 t^{5} d t$

Step 5

Since $- 9$ is constant with respect to $t$ , move $- 9$ out of the integral.

$3 t + C - 12 (\frac{1}{4} t^{4} + C) - 9 \int t^{5} d t$

Step 6

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

$3 t + C - 12 (\frac{1}{4} t^{4} + C) - 9 (\frac{1}{6} t^{6} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$3 t - 3 t^{4} - 9 (\frac{1}{6} t^{6}) + C$

Step 7.2

Simplify.

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Step 7.2.1

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

$3 t - 3 t^{4} - 9 \frac{t^{6}}{6} + C$

Step 7.2.2

Combine $- 9$ and $\frac{t^{6}}{6}$ .

$3 t - 3 t^{4} + \frac{- 9 t^{6}}{6} + C$

Step 7.2.3

Cancel the common factor of $- 9$ and $6$ .

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Step 7.2.3.1

Factor $3$ out of $- 9 t^{6}$ .

$3 t - 3 t^{4} + \frac{3 (- 3 t^{6})}{6} + C$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $3$ out of $6$ .

$3 t - 3 t^{4} + \frac{3 (- 3 t^{6})}{3 (2)} + C$

Step 7.2.3.2.2

Cancel the common factor.

$3 t - 3 t^{4} + \frac{3 (- 3 t^{6})}{3 \cdot 2} + C$

Step 7.2.3.2.3

Rewrite the expression.

$3 t - 3 t^{4} + \frac{- 3 t^{6}}{2} + C$

Step 7.2.4

Move the negative in front of the fraction.

$3 t - 3 t^{4} - \frac{3 t^{6}}{2} + C$

Step 8

Reorder terms.

$3 t - 3 t^{4} - \frac{3}{2} t^{6} + C$