Calculus Examples

$\int_{16}^{20} 3.96 + 0.07 x + 0.015 x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{16}^{20} 3.96 d x + \int_{16}^{20} 0.07 x d x + \int_{16}^{20} 0.015 x^{2} d x$

Step 2

Apply the constant rule.

$3.96 x]_{16}^{20} + \int_{16}^{20} 0.07 x d x + \int_{16}^{20} 0.015 x^{2} d x$

Step 3

Since $0.07$ is constant with respect to $x$ , move $0.07$ out of the integral.

$3.96 x]_{16}^{20} + 0.07 \int_{16}^{20} x d x + \int_{16}^{20} 0.015 x^{2} d x$

Step 4

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

$3.96 x]_{16}^{20} + 0.07 (\frac{1}{2} x^{2}]_{16}^{20}) + \int_{16}^{20} 0.015 x^{2} d x$

Step 5

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

$3.96 x]_{16}^{20} + 0.07 (\frac{x^{2}}{2}]_{16}^{20}) + \int_{16}^{20} 0.015 x^{2} d x$

Step 6

Since $0.015$ is constant with respect to $x$ , move $0.015$ out of the integral.

$3.96 x]_{16}^{20} + 0.07 (\frac{x^{2}}{2}]_{16}^{20}) + 0.015 \int_{16}^{20} x^{2} d x$

Step 7

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

$3.96 x]_{16}^{20} + 0.07 (\frac{x^{2}}{2}]_{16}^{20}) + 0.015 (\frac{1}{3} x^{3}]_{16}^{20})$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

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

$3.96 x]_{16}^{20} + 0.07 (\frac{x^{2}}{2}]_{16}^{20}) + 0.015 (\frac{x^{3}}{3}]_{16}^{20})$

Step 8.2

Substitute and simplify.

Tap for more steps...

Step 8.2.1

Evaluate $3.96 x$ at $20$ and at $16$ .

$(3.96 \cdot 20) - 3.96 \cdot 16 + 0.07 (\frac{x^{2}}{2}]_{16}^{20}) + 0.015 (\frac{x^{3}}{3}]_{16}^{20})$

Step 8.2.2

Evaluate $\frac{x^{2}}{2}$ at $20$ and at $16$ .

$3.96 \cdot 20 - 3.96 \cdot 16 + 0.07 (\frac{20^{2}}{2} - \frac{16^{2}}{2}) + 0.015 (\frac{x^{3}}{3}]_{16}^{20})$

Step 8.2.3

Evaluate $\frac{x^{3}}{3}$ at $20$ and at $16$ .

$3.96 \cdot 20 - 3.96 \cdot 16 + 0.07 (\frac{20^{2}}{2} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4

Simplify.

Tap for more steps...

Step 8.2.4.1

Multiply $3.96$ by $20$ .

$79.2 - 3.96 \cdot 16 + 0.07 (\frac{20^{2}}{2} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.2

Multiply $- 3.96$ by $16$ .

$79.2 - 63.36 + 0.07 (\frac{20^{2}}{2} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.3

Subtract $63.36$ from $79.2$ .

$15.84 + 0.07 (\frac{20^{2}}{2} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.4

Raise $20$ to the power of $2$ .

$15.84 + 0.07 (\frac{400}{2} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.5

Cancel the common factor of $400$ and $2$ .

Tap for more steps...

Step 8.2.4.5.1

Factor $2$ out of $400$ .

$15.84 + 0.07 (\frac{2 \cdot 200}{2} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.5.2

Cancel the common factors.

Tap for more steps...

Step 8.2.4.5.2.1

Factor $2$ out of $2$ .

$15.84 + 0.07 (\frac{2 \cdot 200}{2 (1)} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.5.2.2

Cancel the common factor.

$15.84 + 0.07 (\frac{2 \cdot 200}{2 \cdot 1} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.5.2.3

Rewrite the expression.

$15.84 + 0.07 (\frac{200}{1} - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.5.2.4

Divide $200$ by $1$ .

$15.84 + 0.07 (200 - \frac{16^{2}}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.6

Raise $16$ to the power of $2$ .

$15.84 + 0.07 (200 - \frac{256}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.7

Cancel the common factor of $256$ and $2$ .

Tap for more steps...

Step 8.2.4.7.1

Factor $2$ out of $256$ .

$15.84 + 0.07 (200 - \frac{2 \cdot 128}{2}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.7.2

Cancel the common factors.

Tap for more steps...

Step 8.2.4.7.2.1

Factor $2$ out of $2$ .

$15.84 + 0.07 (200 - \frac{2 \cdot 128}{2 (1)}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.7.2.2

Cancel the common factor.

$15.84 + 0.07 (200 - \frac{2 \cdot 128}{2 \cdot 1}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.7.2.3

Rewrite the expression.

$15.84 + 0.07 (200 - \frac{128}{1}) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.7.2.4

Divide $128$ by $1$ .

$15.84 + 0.07 (200 - 1 \cdot 128) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.8

Multiply $- 1$ by $128$ .

$15.84 + 0.07 (200 - 128) + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.9

Subtract $128$ from $200$ .

$15.84 + 0.07 \cdot 72 + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.10

Multiply $0.07$ by $72$ .

$15.84 + 5.04 + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.11

Add $15.84$ and $5.04$ .

$20.88 + 0.015 ((\frac{20^{3}}{3}) - \frac{16^{3}}{3})$

Step 8.2.4.12

Raise $20$ to the power of $3$ .

$20.88 + 0.015 (\frac{8000}{3} - \frac{16^{3}}{3})$

Step 8.2.4.13

Raise $16$ to the power of $3$ .

$20.88 + 0.015 (\frac{8000}{3} - \frac{4096}{3})$

Step 8.2.4.14

Combine the numerators over the common denominator.

$20.88 + 0.015 \frac{8000 - 4096}{3}$

Step 8.2.4.15

Subtract $4096$ from $8000$ .

$20.88 + 0.015 (\frac{3904}{3})$

Step 8.2.4.16

Combine $0.015$ and $\frac{3904}{3}$ .

$20.88 + \frac{0.015 \cdot 3904}{3}$

Step 8.2.4.17

Multiply $0.015$ by $3904$ .

$20.88 + \frac{58.56}{3}$

Step 8.2.4.18

To write $20.88$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$20.88 \cdot \frac{3}{3} + \frac{58.56}{3}$

Step 8.2.4.19

Combine $20.88$ and $\frac{3}{3}$ .

$\frac{20.88 \cdot 3}{3} + \frac{58.56}{3}$

Step 8.2.4.20

Combine the numerators over the common denominator.

$\frac{20.88 \cdot 3 + 58.56}{3}$

Step 8.2.4.21

Multiply $20.88$ by $3$ .

$\frac{62.64 + 58.56}{3}$

Step 8.2.4.22

Add $62.64$ and $58.56$ .

$\frac{121.2}{3}$

Step 9

Divide $121.2$ by $3$ .

$40.4$

Step 10