Calculus Examples

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

Step 1

Simplify.

Tap for more steps...

Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

$\int 1 \cdot 2 + 1 t^{2} - t (2 + t^{2}) d t$

Step 1.3

Apply the distributive property.

$\int 1 \cdot 2 + 1 t^{2} - t \cdot 2 - t \cdot t^{2} d t$

Step 1.4

Move $t$ .

$\int 1 \cdot 2 + 1 t^{2} - 1 \cdot 2 t - t \cdot t^{2} d t$

Step 1.5

Multiply $2$ by $1$ .

$\int 2 + 1 t^{2} - 1 \cdot 2 t - t \cdot t^{2} d t$

Step 1.6

Multiply $t^{2}$ by $1$ .

$\int 2 + t^{2} - 1 \cdot 2 t - t \cdot t^{2} d t$

Step 1.7

Multiply $- 1$ by $2$ .

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

Step 1.8

Factor out negative.

$\int 2 + t^{2} - 2 t - (t \cdot t^{2}) d t$

Step 1.9

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

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

Step 1.10

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

$\int 2 + t^{2} - 2 t - t^{1 + 2} d t$

Step 1.11

Add $1$ and $2$ .

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

Step 1.12

Reorder $2$ and $t^{2}$ .

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

Step 1.13

Reorder $- 2 t$ and $- t^{3}$ .

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

Step 1.14

Move $2$ .

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

Step 1.15

Reorder $t^{2}$ and $- t^{3}$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

$- (\frac{1}{4} t^{4} + C) + \int t^{2} d t + \int - 2 t d t + \int 2 d t$

Step 5

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

$- (\frac{1}{4} t^{4} + C) + \frac{1}{3} t^{3} + C + \int - 2 t d t + \int 2 d t$

Step 6

Combine $\frac{1}{4}$ and $t^{4}$ .

$- (\frac{t^{4}}{4} + C) + \frac{1}{3} t^{3} + C + \int - 2 t d t + \int 2 d t$

Step 7

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

$- (\frac{t^{4}}{4} + C) + \frac{1}{3} t^{3} + C - 2 \int t d t + \int 2 d t$

Step 8

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

$- (\frac{t^{4}}{4} + C) + \frac{1}{3} t^{3} + C - 2 (\frac{1}{2} t^{2} + C) + \int 2 d t$

Step 9

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

$- (\frac{t^{4}}{4} + C) + \frac{1}{3} t^{3} + C - 2 (\frac{t^{2}}{2} + C) + \int 2 d t$

Step 10

Apply the constant rule.

$- (\frac{t^{4}}{4} + C) + \frac{1}{3} t^{3} + C - 2 (\frac{t^{2}}{2} + C) + 2 t + C$

Step 11

Simplify.

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

Step 12

Reorder terms.

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