Calculus Examples

$\int_{7}^{8} x \sqrt{x - 7} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt{x - 7}$ .

$x (\frac{2}{3} {(x - 7)}^{\frac{3}{2}})]_{7}^{8} - \int_{7}^{8} \frac{2}{3} {(x - 7)}^{\frac{3}{2}} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{2}{3}$ and ${(x - 7)}^{\frac{3}{2}}$ .

$x \frac{2 {(x - 7)}^{\frac{3}{2}}}{3}]_{7}^{8} - \int_{7}^{8} \frac{2}{3} {(x - 7)}^{\frac{3}{2}} d x$

Step 2.2

Combine $x$ and $\frac{2 {(x - 7)}^{\frac{3}{2}}}{3}$ .

$\frac{x (2 {(x - 7)}^{\frac{3}{2}})}{3}]_{7}^{8} - \int_{7}^{8} \frac{2}{3} {(x - 7)}^{\frac{3}{2}} d x$

Step 3

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

$\frac{x (2 {(x - 7)}^{\frac{3}{2}})}{3}]_{7}^{8} - (\frac{2}{3} \int_{7}^{8} {(x - 7)}^{\frac{3}{2}} d x)$

Step 4

Let $u = x - 7$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x - 7$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x - 7$ .

$\frac{d}{d x} [x - 7]$

Step 4.1.2

By the Sum Rule, the derivative of $x - 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 7]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 7]$

Step 4.1.4

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

Substitute the lower limit in for $x$ in $u = x - 7$ .

$u_{lower} = 7 - 7$

Step 4.3

Subtract $7$ from $7$ .

$u_{lower} = 0$

Step 4.4

Substitute the upper limit in for $x$ in $u = x - 7$ .

$u_{upper} = 8 - 7$

Step 4.5

Subtract $7$ from $8$ .

$u_{upper} = 1$

Step 4.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1$

Step 4.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\frac{x (2 {(x - 7)}^{\frac{3}{2}})}{3}]_{7}^{8} - \frac{2}{3} \int_{0}^{1} u^{\frac{3}{2}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$\frac{x (2 {(x - 7)}^{\frac{3}{2}})}{3}]_{7}^{8} - \frac{2}{3} \frac{2}{5} u^{\frac{5}{2}}]_{0}^{1}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{x (2 {(x - 7)}^{\frac{3}{2}})}{3}$ at $8$ and at $7$ .

$(\frac{8 (2 {(8 - 7)}^{\frac{3}{2}})}{3}) - \frac{7 (2 {(7 - 7)}^{\frac{3}{2}})}{3} - \frac{2}{3} \frac{2}{5} u^{\frac{5}{2}}]_{0}^{1}$

Step 6.2

Evaluate $\frac{2}{5} u^{\frac{5}{2}}$ at $1$ and at $0$ .

$(\frac{8 (2 {(8 - 7)}^{\frac{3}{2}})}{3}) - \frac{7 (2 {(7 - 7)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3

Simplify.

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Step 6.3.1

Subtract $7$ from $8$ .

$\frac{8 (2 \cdot 1^{\frac{3}{2}})}{3} - \frac{7 (2 {(7 - 7)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.2

One to any power is one.

$\frac{8 (2 \cdot 1)}{3} - \frac{7 (2 {(7 - 7)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.3

Multiply $2$ by $1$ .

$\frac{8 \cdot 2}{3} - \frac{7 (2 {(7 - 7)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.4

Multiply $8$ by $2$ .

$\frac{16}{3} - \frac{7 (2 {(7 - 7)}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.5

Subtract $7$ from $7$ .

$\frac{16}{3} - \frac{7 (2 \cdot 0^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.6

Rewrite $0$ as $0^{2}$ .

$\frac{16}{3} - \frac{7 (2 \cdot {(0^{2})}^{\frac{3}{2}})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.7

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{16}{3} - \frac{7 (2 \cdot 0^{2 (\frac{3}{2})})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.8

Cancel the common factor of $2$ .

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Step 6.3.8.1

Cancel the common factor.

$\frac{16}{3} - \frac{7 (2 \cdot 0^{2 (\frac{3}{2})})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.8.2

Rewrite the expression.

$\frac{16}{3} - \frac{7 (2 \cdot 0^{3})}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.9

Raising $0$ to any positive power yields $0$ .

$\frac{16}{3} - \frac{7 (2 \cdot 0)}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.10

Multiply $2$ by $0$ .

$\frac{16}{3} - \frac{7 \cdot 0}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.11

Multiply $7$ by $0$ .

$\frac{16}{3} - \frac{0}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.12

Cancel the common factor of $0$ and $3$ .

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Step 6.3.12.1

Factor $3$ out of $0$ .

$\frac{16}{3} - \frac{3 (0)}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.12.2

Cancel the common factors.

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Step 6.3.12.2.1

Factor $3$ out of $3$ .

$\frac{16}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.12.2.2

Cancel the common factor.

$\frac{16}{3} - \frac{3 \cdot 0}{3 \cdot 1} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.12.2.3

Rewrite the expression.

$\frac{16}{3} - \frac{0}{1} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.12.2.4

Divide $0$ by $1$ .

$\frac{16}{3} - 0 - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.13

Multiply $- 1$ by $0$ .

$\frac{16}{3} + 0 - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.14

Add $\frac{16}{3}$ and $0$ .

$\frac{16}{3} - \frac{2}{3} ((\frac{2}{5} \cdot 1^{\frac{5}{2}}) - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.15

One to any power is one.

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} \cdot 1 - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.16

Multiply $\frac{2}{5}$ by $1$ .

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} - \frac{2}{5} \cdot 0^{\frac{5}{2}})$

Step 6.3.17

Rewrite $0$ as $0^{2}$ .

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} - \frac{2}{5} \cdot {(0^{2})}^{\frac{5}{2}})$

Step 6.3.18

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} - \frac{2}{5} \cdot 0^{2 (\frac{5}{2})})$

Step 6.3.19

Cancel the common factor of $2$ .

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Step 6.3.19.1

Cancel the common factor.

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} - \frac{2}{5} \cdot 0^{2 (\frac{5}{2})})$

Step 6.3.19.2

Rewrite the expression.

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} - \frac{2}{5} \cdot 0^{5})$

Step 6.3.20

Raising $0$ to any positive power yields $0$ .

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} - \frac{2}{5} \cdot 0)$

Step 6.3.21

Multiply $0$ by $- 1$ .

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} + 0 (\frac{2}{5}))$

Step 6.3.22

Multiply $0$ by $\frac{2}{5}$ .

$\frac{16}{3} - \frac{2}{3} (\frac{2}{5} + 0)$

Step 6.3.23

Add $\frac{2}{5}$ and $0$ .

$\frac{16}{3} - \frac{2}{3} \cdot \frac{2}{5}$

Step 6.3.24

Multiply $\frac{2}{5}$ by $\frac{2}{3}$ .

$\frac{16}{3} - \frac{2 \cdot 2}{5 \cdot 3}$

Step 6.3.25

Multiply $2$ by $2$ .

$\frac{16}{3} - \frac{4}{5 \cdot 3}$

Step 6.3.26

Multiply $5$ by $3$ .

$\frac{16}{3} - \frac{4}{15}$

Step 6.3.27

To write $\frac{16}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{16}{3} \cdot \frac{5}{5} - \frac{4}{15}$

Step 6.3.28

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.28.1

Multiply $\frac{16}{3}$ by $\frac{5}{5}$ .

$\frac{16 \cdot 5}{3 \cdot 5} - \frac{4}{15}$

Step 6.3.28.2

Multiply $3$ by $5$ .

$\frac{16 \cdot 5}{15} - \frac{4}{15}$

Step 6.3.29

Combine the numerators over the common denominator.

$\frac{16 \cdot 5 - 4}{15}$

Step 6.3.30

Simplify the numerator.

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Step 6.3.30.1

Multiply $16$ by $5$ .

$\frac{80 - 4}{15}$

Step 6.3.30.2

Subtract $4$ from $80$ .

$\frac{76}{15}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{76}{15}$

Decimal Form:

$5.0\overline{6}$

Mixed Number Form:

$5 \frac{1}{15}$

Step 8