Calculus Examples

$\int_{1}^{10} 3 x^{2} + 3 x + 6 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{10} 3 x^{2} d x + \int_{1}^{10} 3 x d x + \int_{1}^{10} 6 d x$

Step 2

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

$3 \int_{1}^{10} x^{2} d x + \int_{1}^{10} 3 x d x + \int_{1}^{10} 6 d x$

Step 3

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

$3 (\frac{1}{3} x^{3}]_{1}^{10}) + \int_{1}^{10} 3 x d x + \int_{1}^{10} 6 d x$

Step 4

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

$3 (\frac{x^{3}}{3}]_{1}^{10}) + \int_{1}^{10} 3 x d x + \int_{1}^{10} 6 d x$

Step 5

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

$3 (\frac{x^{3}}{3}]_{1}^{10}) + 3 \int_{1}^{10} x d x + \int_{1}^{10} 6 d x$

Step 6

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

$3 (\frac{x^{3}}{3}]_{1}^{10}) + 3 (\frac{1}{2} x^{2}]_{1}^{10}) + \int_{1}^{10} 6 d x$

Step 7

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

$3 (\frac{x^{3}}{3}]_{1}^{10}) + 3 (\frac{x^{2}}{2}]_{1}^{10}) + \int_{1}^{10} 6 d x$

Step 8

Apply the constant rule.

$3 (\frac{x^{3}}{3}]_{1}^{10}) + 3 (\frac{x^{2}}{2}]_{1}^{10}) + 6 x]_{1}^{10}$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $\frac{x^{3}}{3}$ at $10$ and at $1$ .

$3 ((\frac{10^{3}}{3}) - \frac{1^{3}}{3}) + 3 (\frac{x^{2}}{2}]_{1}^{10}) + 6 x]_{1}^{10}$

Step 9.2

Evaluate $\frac{x^{2}}{2}$ at $10$ and at $1$ .

$3 (\frac{10^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + 6 x]_{1}^{10}$

Step 9.3

Evaluate $6 x$ at $10$ and at $1$ .

$3 (\frac{10^{3}}{3} - \frac{1^{3}}{3}) + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4

Simplify.

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Step 9.4.1

Raise $10$ to the power of $3$ .

$3 (\frac{1000}{3} - \frac{1^{3}}{3}) + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.2

One to any power is one.

$3 (\frac{1000}{3} - \frac{1}{3}) + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.3

Combine the numerators over the common denominator.

$3 \frac{1000 - 1}{3} + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.4

Subtract $1$ from $1000$ .

$3 (\frac{999}{3}) + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.5

Cancel the common factor of $999$ and $3$ .

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Step 9.4.5.1

Factor $3$ out of $999$ .

$3 \frac{3 \cdot 333}{3} + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.5.2

Cancel the common factors.

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Step 9.4.5.2.1

Factor $3$ out of $3$ .

$3 \frac{3 \cdot 333}{3 (1)} + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.5.2.2

Cancel the common factor.

$3 \frac{3 \cdot 333}{3 \cdot 1} + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.5.2.3

Rewrite the expression.

$3 (\frac{333}{1}) + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.5.2.4

Divide $333$ by $1$ .

$3 \cdot 333 + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.6

Multiply $3$ by $333$ .

$999 + 3 (\frac{10^{2}}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.7

Raise $10$ to the power of $2$ .

$999 + 3 (\frac{100}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.8

Cancel the common factor of $100$ and $2$ .

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Step 9.4.8.1

Factor $2$ out of $100$ .

$999 + 3 (\frac{2 \cdot 50}{2} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.8.2

Cancel the common factors.

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Step 9.4.8.2.1

Factor $2$ out of $2$ .

$999 + 3 (\frac{2 \cdot 50}{2 (1)} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.8.2.2

Cancel the common factor.

$999 + 3 (\frac{2 \cdot 50}{2 \cdot 1} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.8.2.3

Rewrite the expression.

$999 + 3 (\frac{50}{1} - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.8.2.4

Divide $50$ by $1$ .

$999 + 3 (50 - \frac{1^{2}}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.9

One to any power is one.

$999 + 3 (50 - \frac{1}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.10

To write $50$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$999 + 3 (50 \cdot \frac{2}{2} - \frac{1}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.11

Combine $50$ and $\frac{2}{2}$ .

$999 + 3 (\frac{50 \cdot 2}{2} - \frac{1}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.12

Combine the numerators over the common denominator.

$999 + 3 \frac{50 \cdot 2 - 1}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.13

Simplify the numerator.

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Step 9.4.13.1

Multiply $50$ by $2$ .

$999 + 3 \frac{100 - 1}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.13.2

Subtract $1$ from $100$ .

$999 + 3 (\frac{99}{2}) + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.14

Combine $3$ and $\frac{99}{2}$ .

$999 + \frac{3 \cdot 99}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.15

Multiply $3$ by $99$ .

$999 + \frac{297}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.16

To write $999$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$999 \cdot \frac{2}{2} + \frac{297}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.17

Combine $999$ and $\frac{2}{2}$ .

$\frac{999 \cdot 2}{2} + \frac{297}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.18

Combine the numerators over the common denominator.

$\frac{999 \cdot 2 + 297}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.19

Simplify the numerator.

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Step 9.4.19.1

Multiply $999$ by $2$ .

$\frac{1998 + 297}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.19.2

Add $1998$ and $297$ .

$\frac{2295}{2} + (6 \cdot 10) - 6 \cdot 1$

Step 9.4.20

Multiply $6$ by $10$ .

$\frac{2295}{2} + 60 - 6 \cdot 1$

Step 9.4.21

Multiply $- 6$ by $1$ .

$\frac{2295}{2} + 60 - 6$

Step 9.4.22

Subtract $6$ from $60$ .

$\frac{2295}{2} + 54$

Step 9.4.23

To write $54$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2295}{2} + 54 \cdot \frac{2}{2}$

Step 9.4.24

Combine $54$ and $\frac{2}{2}$ .

$\frac{2295}{2} + \frac{54 \cdot 2}{2}$

Step 9.4.25

Combine the numerators over the common denominator.

$\frac{2295 + 54 \cdot 2}{2}$

Step 9.4.26

Simplify the numerator.

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Step 9.4.26.1

Multiply $54$ by $2$ .

$\frac{2295 + 108}{2}$

Step 9.4.26.2

Add $2295$ and $108$ .

$\frac{2403}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2403}{2}$

Decimal Form:

$1201.5$

Mixed Number Form:

$1201 \frac{1}{2}$

Step 11