Calculus Examples

$\int 3 - 2 t - t^{2} d t$

Step 1

Split the single integral into multiple integrals.

$\int 3 d t + \int - 2 t d t + \int - t^{2} d t$

Step 2

Apply the constant rule.

$3 t + C + \int - 2 t d t + \int - t^{2} d t$

Step 3

Since $- 2$ is constant with respect to $t$ , move $- 2$ out of the integral.

$3 t + C - 2 \int t d t + \int - t^{2} d t$

Step 4

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

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

Step 5

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

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

Step 6

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

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify.

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

Step 7.2

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

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

Step 8

Reorder terms.

$3 t - t^{2} - \frac{1}{3} t^{3} + C$