Calculus Examples

$\int 15 t^{2} + 10 t d t$

Step 1

Split the single integral into multiple integrals.

$\int 15 t^{2} d t + \int 10 t d t$

Step 2

Since $15$ is constant with respect to $t$ , move $15$ out of the integral.

$15 \int t^{2} d t + \int 10 t d t$

Step 3

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

$15 (\frac{1}{3} t^{3} + C) + \int 10 t d t$

Step 4

Since $10$ is constant with respect to $t$ , move $10$ out of the integral.

$15 (\frac{1}{3} t^{3} + C) + 10 \int t d t$

Step 5

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

$15 (\frac{1}{3} t^{3} + C) + 10 (\frac{1}{2} t^{2} + C)$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify.

$15 (\frac{1}{3}) t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Combine $15$ and $\frac{1}{3}$ .

$\frac{15}{3} t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2.2

Cancel the common factor of $15$ and $3$ .

Tap for more steps...

Step 6.2.2.1

Factor $3$ out of $15$ .

$\frac{3 \cdot 5}{3} t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 5}{3 (1)} t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2.2.2.2

Cancel the common factor.

$\frac{3 \cdot 5}{3 \cdot 1} t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2.2.2.3

Rewrite the expression.

$\frac{5}{1} t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2.2.2.4

Divide $5$ by $1$ .

$5 t^{3} + 10 (\frac{1}{2}) t^{2} + C$

Step 6.2.3

Combine $10$ and $\frac{1}{2}$ .

$5 t^{3} + \frac{10}{2} t^{2} + C$

Step 6.2.4

Cancel the common factor of $10$ and $2$ .

Tap for more steps...

Step 6.2.4.1

Factor $2$ out of $10$ .

$5 t^{3} + \frac{2 \cdot 5}{2} t^{2} + C$

Step 6.2.4.2

Cancel the common factors.

Tap for more steps...

Step 6.2.4.2.1

Factor $2$ out of $2$ .

$5 t^{3} + \frac{2 \cdot 5}{2 (1)} t^{2} + C$

Step 6.2.4.2.2

Cancel the common factor.

$5 t^{3} + \frac{2 \cdot 5}{2 \cdot 1} t^{2} + C$

Step 6.2.4.2.3

Rewrite the expression.

$5 t^{3} + \frac{5}{1} t^{2} + C$

Step 6.2.4.2.4

Divide $5$ by $1$ .

$5 t^{3} + 5 t^{2} + C$