Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\frac{1}{3} x^{3}]_{1}^{4} + \int_{1}^{4} x d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{3} x^{3}]_{1}^{4}$ and $\frac{1}{2} x^{2}]_{1}^{4}$ .

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{1}{2} x^{2}$ at $4$ and at $1$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Raise $4$ to the power of $3$ .

$\frac{1}{3} \cdot 64 + \frac{1}{2} \cdot 4^{2} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.2

Combine $\frac{1}{3}$ and $64$ .

$\frac{64}{3} + \frac{1}{2} \cdot 4^{2} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.3

Raise $4$ to the power of $2$ .

$\frac{64}{3} + \frac{1}{2} \cdot 16 - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.4

Combine $\frac{1}{2}$ and $16$ .

$\frac{64}{3} + \frac{16}{2} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.5

Cancel the common factor of $16$ and $2$ .

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Step 5.2.2.5.1

Factor $2$ out of $16$ .

$\frac{64}{3} + \frac{2 \cdot 8}{2} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.5.2

Cancel the common factors.

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Step 5.2.2.5.2.1

Factor $2$ out of $2$ .

$\frac{64}{3} + \frac{2 \cdot 8}{2 (1)} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.5.2.2

Cancel the common factor.

$\frac{64}{3} + \frac{2 \cdot 8}{2 \cdot 1} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.5.2.3

Rewrite the expression.

$\frac{64}{3} + \frac{8}{1} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.5.2.4

Divide $8$ by $1$ .

$\frac{64}{3} + 8 - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.6

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{64}{3} + 8 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.7

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

$\frac{64}{3} + \frac{8 \cdot 3}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.8

Combine the numerators over the common denominator.

$\frac{64 + 8 \cdot 3}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.9

Simplify the numerator.

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Step 5.2.2.9.1

Multiply $8$ by $3$ .

$\frac{64 + 24}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.9.2

Add $64$ and $24$ .

$\frac{88}{3} - (\frac{1}{3} \cdot 1^{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.10

One to any power is one.

$\frac{88}{3} - (\frac{1}{3} \cdot 1 + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.11

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

$\frac{88}{3} - (\frac{1}{3} + \frac{1}{2} \cdot 1^{2})$

Step 5.2.2.12

One to any power is one.

$\frac{88}{3} - (\frac{1}{3} + \frac{1}{2} \cdot 1)$

Step 5.2.2.13

Multiply $\frac{1}{2}$ by $1$ .

$\frac{88}{3} - (\frac{1}{3} + \frac{1}{2})$

Step 5.2.2.14

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{88}{3} - (\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2})$

Step 5.2.2.15

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{88}{3} - (\frac{1}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3})$

Step 5.2.2.16

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.16.1

Multiply $\frac{1}{3}$ by $\frac{2}{2}$ .

$\frac{88}{3} - (\frac{2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3})$

Step 5.2.2.16.2

Multiply $3$ by $2$ .

$\frac{88}{3} - (\frac{2}{6} + \frac{1}{2} \cdot \frac{3}{3})$

Step 5.2.2.16.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$\frac{88}{3} - (\frac{2}{6} + \frac{3}{2 \cdot 3})$

Step 5.2.2.16.4

Multiply $2$ by $3$ .

$\frac{88}{3} - (\frac{2}{6} + \frac{3}{6})$

Step 5.2.2.17

Combine the numerators over the common denominator.

$\frac{88}{3} - \frac{2 + 3}{6}$

Step 5.2.2.18

Add $2$ and $3$ .

$\frac{88}{3} - \frac{5}{6}$

Step 5.2.2.19

To write $\frac{88}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{88}{3} \cdot \frac{2}{2} - \frac{5}{6}$

Step 5.2.2.20

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.2.20.1

Multiply $\frac{88}{3}$ by $\frac{2}{2}$ .

$\frac{88 \cdot 2}{3 \cdot 2} - \frac{5}{6}$

Step 5.2.2.20.2

Multiply $3$ by $2$ .

$\frac{88 \cdot 2}{6} - \frac{5}{6}$

Step 5.2.2.21

Combine the numerators over the common denominator.

$\frac{88 \cdot 2 - 5}{6}$

Step 5.2.2.22

Simplify the numerator.

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Step 5.2.2.22.1

Multiply $88$ by $2$ .

$\frac{176 - 5}{6}$

Step 5.2.2.22.2

Subtract $5$ from $176$ .

$\frac{171}{6}$

Step 5.2.2.23

Cancel the common factor of $171$ and $6$ .

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Step 5.2.2.23.1

Factor $3$ out of $171$ .

$\frac{3 (57)}{6}$

Step 5.2.2.23.2

Cancel the common factors.

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Step 5.2.2.23.2.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 57}{3 \cdot 2}$

Step 5.2.2.23.2.2

Cancel the common factor.

$\frac{3 \cdot 57}{3 \cdot 2}$

Step 5.2.2.23.2.3

Rewrite the expression.

$\frac{57}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{57}{2}$

Decimal Form:

$28.5$

Mixed Number Form:

$28 \frac{1}{2}$

Step 7