Calculus Examples

$\int - 9 t^{2} + 14 t - 2 d t$

Step 1

Split the single integral into multiple integrals.

$\int - 9 t^{2} d t + \int 14 t d t + \int - 2 d t$

Step 2

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

$- 9 \int t^{2} d t + \int 14 t d t + \int - 2 d t$

Step 3

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

$- 9 (\frac{1}{3} t^{3} + C) + \int 14 t d t + \int - 2 d t$

Step 4

Since $14$ is constant with respect to $t$ , move $14$ out of the integral.

$- 9 (\frac{1}{3} t^{3} + C) + 14 \int t d t + \int - 2 d t$

Step 5

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

$- 9 (\frac{1}{3} t^{3} + C) + 14 (\frac{1}{2} t^{2} + C) + \int - 2 d t$

Step 6

Apply the constant rule.

$- 9 (\frac{1}{3} t^{3} + C) + 14 (\frac{1}{2} t^{2} + C) - 2 t + C$

Step 7

Simplify.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{3}$ and $t^{3}$ .

$- 9 (\frac{t^{3}}{3} + C) + 14 (\frac{1}{2} t^{2} + C) - 2 t + C$

Step 7.1.2

Combine $\frac{1}{2}$ and $t^{2}$ .

$- 9 (\frac{t^{3}}{3} + C) + 14 (\frac{t^{2}}{2} + C) - 2 t + C$

Step 7.2

Simplify.

$- 3 t^{3} + 7 t^{2} - 2 t + C$