Calculus Examples

$\int_{0}^{7} x \sqrt[3]{x + 1} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt[3]{x + 1}$ .

$x (\frac{3}{4} {(x + 1)}^{\frac{4}{3}})]_{0}^{7} - \int_{0}^{7} \frac{3}{4} {(x + 1)}^{\frac{4}{3}} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{3}{4}$ and ${(x + 1)}^{\frac{4}{3}}$ .

$x \frac{3 {(x + 1)}^{\frac{4}{3}}}{4}]_{0}^{7} - \int_{0}^{7} \frac{3}{4} {(x + 1)}^{\frac{4}{3}} d x$

Step 2.2

Combine $x$ and $\frac{3 {(x + 1)}^{\frac{4}{3}}}{4}$ .

$\frac{x (3 {(x + 1)}^{\frac{4}{3}})}{4}]_{0}^{7} - \int_{0}^{7} \frac{3}{4} {(x + 1)}^{\frac{4}{3}} d x$

Step 3

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

$\frac{x (3 {(x + 1)}^{\frac{4}{3}})}{4}]_{0}^{7} - (\frac{3}{4} \int_{0}^{7} {(x + 1)}^{\frac{4}{3}} d x)$

Step 4

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x + 1$ .

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

Step 4.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 4.1.3

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

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

Step 4.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ 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 $u = x + 1$ .

$u_{lower} = 0 + 1$

Step 4.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 4.4

Substitute the upper limit in for $x$ in $u = x + 1$ .

$u_{upper} = 7 + 1$

Step 4.5

Add $7$ and $1$ .

$u_{upper} = 8$

Step 4.6

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

$u_{lower} = 1$

$u_{upper} = 8$

Step 4.7

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

$\frac{x (3 {(x + 1)}^{\frac{4}{3}})}{4}]_{0}^{7} - \frac{3}{4} \int_{1}^{8} u^{\frac{4}{3}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{4}{3}}$ with respect to $u$ is $\frac{3}{7} u^{\frac{7}{3}}$ .

$\frac{x (3 {(x + 1)}^{\frac{4}{3}})}{4}]_{0}^{7} - \frac{3}{4} \frac{3}{7} u^{\frac{7}{3}}]_{1}^{8}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{x (3 {(x + 1)}^{\frac{4}{3}})}{4}$ at $7$ and at $0$ .

$(\frac{7 (3 {(7 + 1)}^{\frac{4}{3}})}{4}) - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} \frac{3}{7} u^{\frac{7}{3}}]_{1}^{8}$

Step 6.2

Evaluate $\frac{3}{7} u^{\frac{7}{3}}$ at $8$ and at $1$ .

$(\frac{7 (3 {(7 + 1)}^{\frac{4}{3}})}{4}) - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3

Simplify.

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Step 6.3.1

Add $7$ and $1$ .

$\frac{7 (3 \cdot 8^{\frac{4}{3}})}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.2

Rewrite $8$ as $2^{3}$ .

$\frac{7 (3 \cdot {(2^{3})}^{\frac{4}{3}})}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.3

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

$\frac{7 (3 \cdot 2^{3 (\frac{4}{3})})}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.4

Cancel the common factor of $3$ .

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Step 6.3.4.1

Cancel the common factor.

$\frac{7 (3 \cdot 2^{3 (\frac{4}{3})})}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.4.2

Rewrite the expression.

$\frac{7 (3 \cdot 2^{4})}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.5

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

$\frac{7 (3 \cdot 16)}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.6

Multiply $3$ by $16$ .

$\frac{7 \cdot 48}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.7

Multiply $7$ by $48$ .

$\frac{336}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.8

Cancel the common factor of $336$ and $4$ .

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Step 6.3.8.1

Factor $4$ out of $336$ .

$\frac{4 \cdot 84}{4} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.8.2

Cancel the common factors.

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Step 6.3.8.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 84}{4 (1)} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.8.2.2

Cancel the common factor.

$\frac{4 \cdot 84}{4 \cdot 1} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.8.2.3

Rewrite the expression.

$\frac{84}{1} - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.8.2.4

Divide $84$ by $1$ .

$84 - \frac{0 (3 {(0 + 1)}^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.9

Add $0$ and $1$ .

$84 - \frac{0 (3 \cdot 1^{\frac{4}{3}})}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.10

One to any power is one.

$84 - \frac{0 (3 \cdot 1)}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.11

Multiply $3$ by $1$ .

$84 - \frac{0 \cdot 3}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.12

Multiply $0$ by $3$ .

$84 - \frac{0}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.13

Cancel the common factor of $0$ and $4$ .

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Step 6.3.13.1

Factor $4$ out of $0$ .

$84 - \frac{4 (0)}{4} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.13.2

Cancel the common factors.

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Step 6.3.13.2.1

Factor $4$ out of $4$ .

$84 - \frac{4 \cdot 0}{4 \cdot 1} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.13.2.2

Cancel the common factor.

$84 - \frac{4 \cdot 0}{4 \cdot 1} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.13.2.3

Rewrite the expression.

$84 - \frac{0}{1} - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.13.2.4

Divide $0$ by $1$ .

$84 - 0 - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.14

Multiply $- 1$ by $0$ .

$84 + 0 - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.15

Add $84$ and $0$ .

$84 - \frac{3}{4} ((\frac{3}{7} \cdot 8^{\frac{7}{3}}) - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.16

Rewrite $8$ as $2^{3}$ .

$84 - \frac{3}{4} (\frac{3}{7} \cdot {(2^{3})}^{\frac{7}{3}} - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.17

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

$84 - \frac{3}{4} (\frac{3}{7} \cdot 2^{3 (\frac{7}{3})} - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.18

Cancel the common factor of $3$ .

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Step 6.3.18.1

Cancel the common factor.

$84 - \frac{3}{4} (\frac{3}{7} \cdot 2^{3 (\frac{7}{3})} - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.18.2

Rewrite the expression.

$84 - \frac{3}{4} (\frac{3}{7} \cdot 2^{7} - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.19

Raise $2$ to the power of $7$ .

$84 - \frac{3}{4} (\frac{3}{7} \cdot 128 - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.20

Combine $\frac{3}{7}$ and $128$ .

$84 - \frac{3}{4} (\frac{3 \cdot 128}{7} - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.21

Multiply $3$ by $128$ .

$84 - \frac{3}{4} (\frac{384}{7} - \frac{3}{7} \cdot 1^{\frac{7}{3}})$

Step 6.3.22

One to any power is one.

$84 - \frac{3}{4} (\frac{384}{7} - \frac{3}{7} \cdot 1)$

Step 6.3.23

Multiply $- 1$ by $1$ .

$84 - \frac{3}{4} (\frac{384}{7} - \frac{3}{7})$

Step 6.3.24

Combine the numerators over the common denominator.

$84 - \frac{3}{4} \cdot \frac{384 - 3}{7}$

Step 6.3.25

Subtract $3$ from $384$ .

$84 - \frac{3}{4} \cdot \frac{381}{7}$

Step 6.3.26

Multiply $\frac{381}{7}$ by $\frac{3}{4}$ .

$84 - \frac{381 \cdot 3}{7 \cdot 4}$

Step 6.3.27

Multiply $381$ by $3$ .

$84 - \frac{1143}{7 \cdot 4}$

Step 6.3.28

Multiply $7$ by $4$ .

$84 - \frac{1143}{28}$

Step 6.3.29

To write $84$ as a fraction with a common denominator, multiply by $\frac{28}{28}$ .

$84 \cdot \frac{28}{28} - \frac{1143}{28}$

Step 6.3.30

Combine $84$ and $\frac{28}{28}$ .

$\frac{84 \cdot 28}{28} - \frac{1143}{28}$

Step 6.3.31

Combine the numerators over the common denominator.

$\frac{84 \cdot 28 - 1143}{28}$

Step 6.3.32

Simplify the numerator.

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Step 6.3.32.1

Multiply $84$ by $28$ .

$\frac{2352 - 1143}{28}$

Step 6.3.32.2

Subtract $1143$ from $2352$ .

$\frac{1209}{28}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1209}{28}$

Decimal Form:

$43.17857142 \dots$

Mixed Number Form:

$43 \frac{5}{28}$

Step 8