Calculus Examples

$\int_{0}^{- 3} x^{2} - 15 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{- 3} x^{2} d x + \int_{0}^{- 3} - 15 d x$

Step 2

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

$\frac{1}{3} x^{3}]_{0}^{- 3} + \int_{0}^{- 3} - 15 d x$

Step 3

Apply the constant rule.

$\frac{1}{3} x^{3}]_{0}^{- 3} + - 15 x]_{0}^{- 3}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Combine $\frac{1}{3} x^{3}]_{0}^{- 3}$ and $- 15 x]_{0}^{- 3}$ .

$\frac{1}{3} x^{3} - 15 x]_{0}^{- 3}$

Step 4.2

Substitute and simplify.

Tap for more steps...

Step 4.2.1

Evaluate $\frac{1}{3} x^{3} - 15 x$ at $- 3$ and at $0$ .

$(\frac{1}{3} {(- 3)}^{3} - 15 \cdot - 3) - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Raise $- 3$ to the power of $3$ .

$\frac{1}{3} \cdot - 27 - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.2

Combine $\frac{1}{3}$ and $- 27$ .

$\frac{- 27}{3} - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.3

Cancel the common factor of $- 27$ and $3$ .

Tap for more steps...

Step 4.2.2.3.1

Factor $3$ out of $- 27$ .

$\frac{3 \cdot - 9}{3} - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.3.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 9}{3 (1)} - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 9}{3 \cdot 1} - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.3.2.3

Rewrite the expression.

$\frac{- 9}{1} - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.3.2.4

Divide $- 9$ by $1$ .

$- 9 - 15 \cdot - 3 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.4

Multiply $- 15$ by $- 3$ .

$- 9 + 45 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.5

Add $- 9$ and $45$ .

$36 - (\frac{1}{3} \cdot 0^{3} - 15 \cdot 0)$

Step 4.2.2.6

Raising $0$ to any positive power yields $0$ .

$36 - (\frac{1}{3} \cdot 0 - 15 \cdot 0)$

Step 4.2.2.7

Multiply $\frac{1}{3}$ by $0$ .

$36 - (0 - 15 \cdot 0)$

Step 4.2.2.8

Multiply $- 15$ by $0$ .

$36 - (0 + 0)$

Step 4.2.2.9

Add $0$ and $0$ .

$36 - 0$

Step 4.2.2.10

Multiply $- 1$ by $0$ .

$36 + 0$

Step 4.2.2.11

Add $36$ and $0$ .

$36$

Step 5