Calculus Examples

$\int_{1}^{6} 3 x^{2} - 4 x - 5 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{6} 3 x^{2} d x + \int_{1}^{6} - 4 x d x + \int_{1}^{6} - 5 d x$

Step 2

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

$3 \int_{1}^{6} x^{2} d x + \int_{1}^{6} - 4 x d x + \int_{1}^{6} - 5 d x$

Step 3

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

$3 (\frac{1}{3} x^{3}]_{1}^{6}) + \int_{1}^{6} - 4 x d x + \int_{1}^{6} - 5 d x$

Step 4

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

$3 (\frac{x^{3}}{3}]_{1}^{6}) + \int_{1}^{6} - 4 x d x + \int_{1}^{6} - 5 d x$

Step 5

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$3 (\frac{x^{3}}{3}]_{1}^{6}) - 4 \int_{1}^{6} x d x + \int_{1}^{6} - 5 d x$

Step 6

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

$3 (\frac{x^{3}}{3}]_{1}^{6}) - 4 (\frac{1}{2} x^{2}]_{1}^{6}) + \int_{1}^{6} - 5 d x$

Step 7

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

$3 (\frac{x^{3}}{3}]_{1}^{6}) - 4 (\frac{x^{2}}{2}]_{1}^{6}) + \int_{1}^{6} - 5 d x$

Step 8

Apply the constant rule.

$3 (\frac{x^{3}}{3}]_{1}^{6}) - 4 (\frac{x^{2}}{2}]_{1}^{6}) + - 5 x]_{1}^{6}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{3}}{3}$ at $6$ and at $1$ .

$3 ((\frac{6^{3}}{3}) - \frac{1^{3}}{3}) - 4 (\frac{x^{2}}{2}]_{1}^{6}) + - 5 x]_{1}^{6}$

Step 9.2

Evaluate $\frac{x^{2}}{2}$ at $6$ and at $1$ .

$3 (\frac{6^{3}}{3} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + - 5 x]_{1}^{6}$

Step 9.3

Evaluate $- 5 x$ at $6$ and at $1$ .

$3 (\frac{6^{3}}{3} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4

Simplify.

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Step 9.4.1

Raise $6$ to the power of $3$ .

$3 (\frac{216}{3} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.2

Cancel the common factor of $216$ and $3$ .

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Step 9.4.2.1

Factor $3$ out of $216$ .

$3 (\frac{3 \cdot 72}{3} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.2.2

Cancel the common factors.

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Step 9.4.2.2.1

Factor $3$ out of $3$ .

$3 (\frac{3 \cdot 72}{3 (1)} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.2.2.2

Cancel the common factor.

$3 (\frac{3 \cdot 72}{3 \cdot 1} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.2.2.3

Rewrite the expression.

$3 (\frac{72}{1} - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.2.2.4

Divide $72$ by $1$ .

$3 (72 - \frac{1^{3}}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.3

One to any power is one.

$3 (72 - \frac{1}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.4

To write $72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 (72 \cdot \frac{3}{3} - \frac{1}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.5

Combine $72$ and $\frac{3}{3}$ .

$3 (\frac{72 \cdot 3}{3} - \frac{1}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.6

Combine the numerators over the common denominator.

$3 \frac{72 \cdot 3 - 1}{3} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.7

Simplify the numerator.

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Step 9.4.7.1

Multiply $72$ by $3$ .

$3 \frac{216 - 1}{3} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.7.2

Subtract $1$ from $216$ .

$3 (\frac{215}{3}) - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.8

Combine $3$ and $\frac{215}{3}$ .

$\frac{3 \cdot 215}{3} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.9

Multiply $3$ by $215$ .

$\frac{645}{3} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.10

Cancel the common factor of $645$ and $3$ .

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Step 9.4.10.1

Factor $3$ out of $645$ .

$\frac{3 \cdot 215}{3} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.10.2

Cancel the common factors.

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Step 9.4.10.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 215}{3 (1)} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.10.2.2

Cancel the common factor.

$\frac{3 \cdot 215}{3 \cdot 1} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.10.2.3

Rewrite the expression.

$\frac{215}{1} - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.10.2.4

Divide $215$ by $1$ .

$215 - 4 (\frac{6^{2}}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.11

Raise $6$ to the power of $2$ .

$215 - 4 (\frac{36}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.12

Cancel the common factor of $36$ and $2$ .

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Step 9.4.12.1

Factor $2$ out of $36$ .

$215 - 4 (\frac{2 \cdot 18}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.12.2

Cancel the common factors.

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Step 9.4.12.2.1

Factor $2$ out of $2$ .

$215 - 4 (\frac{2 \cdot 18}{2 (1)} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.12.2.2

Cancel the common factor.

$215 - 4 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.12.2.3

Rewrite the expression.

$215 - 4 (\frac{18}{1} - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.12.2.4

Divide $18$ by $1$ .

$215 - 4 (18 - \frac{1^{2}}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.13

One to any power is one.

$215 - 4 (18 - \frac{1}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.14

To write $18$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$215 - 4 (18 \cdot \frac{2}{2} - \frac{1}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.15

Combine $18$ and $\frac{2}{2}$ .

$215 - 4 (\frac{18 \cdot 2}{2} - \frac{1}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.16

Combine the numerators over the common denominator.

$215 - 4 \frac{18 \cdot 2 - 1}{2} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.17

Simplify the numerator.

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Step 9.4.17.1

Multiply $18$ by $2$ .

$215 - 4 \frac{36 - 1}{2} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.17.2

Subtract $1$ from $36$ .

$215 - 4 (\frac{35}{2}) + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.18

Combine $- 4$ and $\frac{35}{2}$ .

$215 + \frac{- 4 \cdot 35}{2} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.19

Multiply $- 4$ by $35$ .

$215 + \frac{- 140}{2} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.20

Cancel the common factor of $- 140$ and $2$ .

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Step 9.4.20.1

Factor $2$ out of $- 140$ .

$215 + \frac{2 \cdot - 70}{2} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.20.2

Cancel the common factors.

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Step 9.4.20.2.1

Factor $2$ out of $2$ .

$215 + \frac{2 \cdot - 70}{2 (1)} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.20.2.2

Cancel the common factor.

$215 + \frac{2 \cdot - 70}{2 \cdot 1} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.20.2.3

Rewrite the expression.

$215 + \frac{- 70}{1} + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.20.2.4

Divide $- 70$ by $1$ .

$215 - 70 + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.21

Subtract $70$ from $215$ .

$145 + (- 5 \cdot 6) + 5 \cdot 1$

Step 9.4.22

Multiply $- 5$ by $6$ .

$145 - 30 + 5 \cdot 1$

Step 9.4.23

Multiply $5$ by $1$ .

$145 - 30 + 5$

Step 9.4.24

Add $- 30$ and $5$ .

$145 - 25$

Step 9.4.25

Subtract $25$ from $145$ .

$120$

Step 10