Calculus Examples

$\int_{- 1}^{2} - x^{2} + 4 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 1}^{2} - x^{2} d x + \int_{- 1}^{2} 4 d x$

Step 2

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

$- \int_{- 1}^{2} x^{2} d x + \int_{- 1}^{2} 4 d x$

Step 3

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

$- (\frac{1}{3} x^{3}]_{- 1}^{2}) + \int_{- 1}^{2} 4 d x$

Step 4

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

$- (\frac{x^{3}}{3}]_{- 1}^{2}) + \int_{- 1}^{2} 4 d x$

Step 5

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{- 1}^{2}) + 4 x]_{- 1}^{2}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $- 1$ .

$- ((\frac{2^{3}}{3}) - \frac{{(- 1)}^{3}}{3}) + 4 x]_{- 1}^{2}$

Step 6.2

Evaluate $4 x$ at $2$ and at $- 1$ .

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

Step 6.3

Simplify.

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Step 6.3.1

Raise $2$ to the power of $3$ .

$- (\frac{8}{3} - \frac{{(- 1)}^{3}}{3}) + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.2

Raise $- 1$ to the power of $3$ .

$- (\frac{8}{3} - \frac{- 1}{3}) + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.3

Move the negative in front of the fraction.

$- (\frac{8}{3} - - \frac{1}{3}) + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.4

Multiply $- 1$ by $- 1$ .

$- (\frac{8}{3} + 1 (\frac{1}{3})) + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.5

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

$- (\frac{8}{3} + \frac{1}{3}) + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.6

Combine the numerators over the common denominator.

$- \frac{8 + 1}{3} + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.7

Add $8$ and $1$ .

$- \frac{9}{3} + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.8

Cancel the common factor of $9$ and $3$ .

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Step 6.3.8.1

Factor $3$ out of $9$ .

$- \frac{3 \cdot 3}{3} + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.8.2

Cancel the common factors.

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Step 6.3.8.2.1

Factor $3$ out of $3$ .

$- \frac{3 \cdot 3}{3 (1)} + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.8.2.2

Cancel the common factor.

$- \frac{3 \cdot 3}{3 \cdot 1} + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.8.2.3

Rewrite the expression.

$- \frac{3}{1} + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.8.2.4

Divide $3$ by $1$ .

$- 1 \cdot 3 + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.9

Multiply $- 1$ by $3$ .

$- 3 + 4 \cdot 2 - 4 \cdot - 1$

Step 6.3.10

Multiply $4$ by $2$ .

$- 3 + 8 - 4 \cdot - 1$

Step 6.3.11

Multiply $- 4$ by $- 1$ .

$- 3 + 8 + 4$

Step 6.3.12

Add $8$ and $4$ .

$- 3 + 12$

Step 6.3.13

Add $- 3$ and $12$ .

$9$

Step 7