Calculus Examples

$\int_{1}^{2} x^{2} - 3 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{2} x^{2} d x + \int_{1}^{2} - 3 d x$

Step 2

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

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

Step 3

Apply the constant rule.

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

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{3} x^{3}]_{1}^{2}$ and $- 3 x]_{1}^{2}$ .

$\frac{1}{3} x^{3} - 3 x]_{1}^{2}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

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

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Combine $\frac{1}{3}$ and $8$ .

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

Step 4.2.2.3

Multiply $- 3$ by $2$ .

$\frac{8}{3} - 6 - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.4

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} - 6 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.5

Combine $- 6$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{- 6 \cdot 3}{3} - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.6

Combine the numerators over the common denominator.

$\frac{8 - 6 \cdot 3}{3} - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.7

Simplify the numerator.

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Step 4.2.2.7.1

Multiply $- 6$ by $3$ .

$\frac{8 - 18}{3} - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.7.2

Subtract $18$ from $8$ .

$\frac{- 10}{3} - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.8

Move the negative in front of the fraction.

$- \frac{10}{3} - (\frac{1}{3} \cdot 1^{3} - 3 \cdot 1)$

Step 4.2.2.9

One to any power is one.

$- \frac{10}{3} - (\frac{1}{3} \cdot 1 - 3 \cdot 1)$

Step 4.2.2.10

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

$- \frac{10}{3} - (\frac{1}{3} - 3 \cdot 1)$

Step 4.2.2.11

Multiply $- 3$ by $1$ .

$- \frac{10}{3} - (\frac{1}{3} - 3)$

Step 4.2.2.12

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{10}{3} - (\frac{1}{3} - 3 \cdot \frac{3}{3})$

Step 4.2.2.13

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

$- \frac{10}{3} - (\frac{1}{3} + \frac{- 3 \cdot 3}{3})$

Step 4.2.2.14

Combine the numerators over the common denominator.

$- \frac{10}{3} - \frac{1 - 3 \cdot 3}{3}$

Step 4.2.2.15

Simplify the numerator.

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Step 4.2.2.15.1

Multiply $- 3$ by $3$ .

$- \frac{10}{3} - \frac{1 - 9}{3}$

Step 4.2.2.15.2

Subtract $9$ from $1$ .

$- \frac{10}{3} - \frac{- 8}{3}$

Step 4.2.2.16

Move the negative in front of the fraction.

$- \frac{10}{3} - - \frac{8}{3}$

Step 4.2.2.17

Multiply $- 1$ by $- 1$ .

$- \frac{10}{3} + 1 (\frac{8}{3})$

Step 4.2.2.18

Multiply $\frac{8}{3}$ by $1$ .

$- \frac{10}{3} + \frac{8}{3}$

Step 4.2.2.19

Combine the numerators over the common denominator.

$\frac{- 10 + 8}{3}$

Step 4.2.2.20

Add $- 10$ and $8$ .

$\frac{- 2}{3}$

Step 4.2.2.21

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3}$

Decimal Form:

$- 0.\overline{6}$

Step 6