Calculus Examples

$\int 3 t^{2} + \frac{t}{9} d t$

Step 1

Split the single integral into multiple integrals.

$\int 3 t^{2} d t + \int \frac{t}{9} d t$

Step 2

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int t^{2} d t + \int \frac{t}{9} d t$

Step 3

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

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

Step 4

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

$3 (\frac{t^{3}}{3} + C) + \int \frac{t}{9} d t$

Step 5

Since $\frac{1}{9}$ is constant with respect to $t$ , move $\frac{1}{9}$ out of the integral.

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

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

$t^{3} + \frac{1}{9} \cdot \frac{t^{2}}{2} + C$

Step 7.2.2

Multiply $\frac{1}{9}$ by $\frac{t^{2}}{2}$ .

$t^{3} + \frac{t^{2}}{9 \cdot 2} + C$

Step 7.2.3

Multiply $9$ by $2$ .

$t^{3} + \frac{t^{2}}{18} + C$

Step 7.3

Reorder terms.

$t^{3} + \frac{1}{18} t^{2} + C$