Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Substitute and simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Simplify.

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Step 9.4.1

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

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

Step 9.4.2

One to any power is one.

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

Step 9.4.3

Combine the numerators over the common denominator.

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

Step 9.4.4

Subtract $1$ from $8$ .

$3 (\frac{7}{3}) - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.5

Combine $3$ and $\frac{7}{3}$ .

$\frac{3 \cdot 7}{3} - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.6

Multiply $3$ by $7$ .

$\frac{21}{3} - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.7

Cancel the common factor of $21$ and $3$ .

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Step 9.4.7.1

Factor $3$ out of $21$ .

$\frac{3 \cdot 7}{3} - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.7.2

Cancel the common factors.

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Step 9.4.7.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 7}{3 (1)} - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.7.2.2

Cancel the common factor.

$\frac{3 \cdot 7}{3 \cdot 1} - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.7.2.3

Rewrite the expression.

$\frac{7}{1} - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.7.2.4

Divide $7$ by $1$ .

$7 - 2 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.8

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

$7 - 2 (\frac{4}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.9

Cancel the common factor of $4$ and $2$ .

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Step 9.4.9.1

Factor $2$ out of $4$ .

$7 - 2 (\frac{2 \cdot 2}{2} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.9.2

Cancel the common factors.

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Step 9.4.9.2.1

Factor $2$ out of $2$ .

$7 - 2 (\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.9.2.2

Cancel the common factor.

$7 - 2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.9.2.3

Rewrite the expression.

$7 - 2 (\frac{2}{1} - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.9.2.4

Divide $2$ by $1$ .

$7 - 2 (2 - \frac{1^{2}}{2}) + (2) - 1$

Step 9.4.10

One to any power is one.

$7 - 2 (2 - \frac{1}{2}) + (2) - 1$

Step 9.4.11

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 - 2 (2 \cdot \frac{2}{2} - \frac{1}{2}) + (2) - 1$

Step 9.4.12

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

$7 - 2 (\frac{2 \cdot 2}{2} - \frac{1}{2}) + (2) - 1$

Step 9.4.13

Combine the numerators over the common denominator.

$7 - 2 \frac{2 \cdot 2 - 1}{2} + (2) - 1$

Step 9.4.14

Simplify the numerator.

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Step 9.4.14.1

Multiply $2$ by $2$ .

$7 - 2 \frac{4 - 1}{2} + (2) - 1$

Step 9.4.14.2

Subtract $1$ from $4$ .

$7 - 2 (\frac{3}{2}) + (2) - 1$

Step 9.4.15

Combine $- 2$ and $\frac{3}{2}$ .

$7 + \frac{- 2 \cdot 3}{2} + (2) - 1$

Step 9.4.16

Multiply $- 2$ by $3$ .

$7 + \frac{- 6}{2} + (2) - 1$

Step 9.4.17

Cancel the common factor of $- 6$ and $2$ .

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Step 9.4.17.1

Factor $2$ out of $- 6$ .

$7 + \frac{2 \cdot - 3}{2} + (2) - 1$

Step 9.4.17.2

Cancel the common factors.

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Step 9.4.17.2.1

Factor $2$ out of $2$ .

$7 + \frac{2 \cdot - 3}{2 (1)} + (2) - 1$

Step 9.4.17.2.2

Cancel the common factor.

$7 + \frac{2 \cdot - 3}{2 \cdot 1} + (2) - 1$

Step 9.4.17.2.3

Rewrite the expression.

$7 + \frac{- 3}{1} + (2) - 1$

Step 9.4.17.2.4

Divide $- 3$ by $1$ .

$7 - 3 + (2) - 1$

Step 9.4.18

Subtract $3$ from $7$ .

$4 + (2) - 1$

Step 9.4.19

Subtract $1$ from $2$ .

$4 + 1$

Step 9.4.20

Add $4$ and $1$ .

$5$

Step 10