Calculus Examples

$\int_{2}^{7} x \sqrt{x + 2} d x$

Step 1

Let $u = x + 2$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x + 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x + 2$ .

$\frac{d}{d x} [x + 2]$

Step 1.1.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 1.1.3

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

$1 + \frac{d}{d x} [2]$

Step 1.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $x$ in $u = x + 2$ .

$u_{lower} = 2 + 2$

Step 1.3

Add $2$ and $2$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = x + 2$ .

$u_{upper} = 7 + 2$

Step 1.5

Add $7$ and $2$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 4$

$u_{upper} = 9$

Step 1.7

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

$\int_{4}^{9} (u - 2) \sqrt{u} d u$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int_{4}^{9} (u - 2) u^{\frac{1}{2}} d u$

Step 3

Expand $(u - 2) u^{\frac{1}{2}}$ .

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Step 3.1

Apply the distributive property.

$\int_{4}^{9} u \cdot u^{\frac{1}{2}} - 2 u^{\frac{1}{2}} d u$

Step 3.2

Raise $u$ to the power of $1$ .

$\int_{4}^{9} u^{1} u^{\frac{1}{2}} - 2 u^{\frac{1}{2}} d u$

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{4}^{9} u^{1 + \frac{1}{2}} - 2 u^{\frac{1}{2}} d u$

Step 3.4

Write $1$ as a fraction with a common denominator.

$\int_{4}^{9} u^{\frac{2}{2} + \frac{1}{2}} - 2 u^{\frac{1}{2}} d u$

Step 3.5

Combine the numerators over the common denominator.

$\int_{4}^{9} u^{\frac{2 + 1}{2}} - 2 u^{\frac{1}{2}} d u$

Step 3.6

Add $2$ and $1$ .

$\int_{4}^{9} u^{\frac{3}{2}} - 2 u^{\frac{1}{2}} d u$

Step 4

Split the single integral into multiple integrals.

$\int_{4}^{9} u^{\frac{3}{2}} d u + \int_{4}^{9} - 2 u^{\frac{1}{2}} d u$

Step 5

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

$\frac{2}{5} u^{\frac{5}{2}}]_{4}^{9} + \int_{4}^{9} - 2 u^{\frac{1}{2}} d u$

Step 6

Since $- 2$ is constant with respect to $u$ , move $- 2$ out of the integral.

$\frac{2}{5} u^{\frac{5}{2}}]_{4}^{9} - 2 \int_{4}^{9} u^{\frac{1}{2}} d u$

Step 7

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

$\frac{2}{5} u^{\frac{5}{2}}]_{4}^{9} - 2 (\frac{2}{3} u^{\frac{3}{2}}]_{4}^{9})$

Step 8

Simplify the expression.

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Step 8.1

Evaluate $\frac{2}{5} u^{\frac{5}{2}}$ at $9$ and at $4$ .

$(\frac{2}{5} \cdot 9^{\frac{5}{2}}) - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} u^{\frac{3}{2}}]_{4}^{9})$

Step 8.2

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $4$ .

$\frac{2}{5} \cdot 9^{\frac{5}{2}} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.3

Simplify.

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Step 8.3.1

Rewrite $9$ as $3^{2}$ .

$\frac{2}{5} \cdot {(3^{2})}^{\frac{5}{2}} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.3.2

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

$\frac{2}{5} \cdot 3^{2 (\frac{5}{2})} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.3.3

Cancel the common factor of $2$ .

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Step 8.3.3.1

Cancel the common factor.

$\frac{2}{5} \cdot 3^{2 (\frac{5}{2})} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.3.3.2

Rewrite the expression.

$\frac{2}{5} \cdot 3^{5} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.3.4

Raise $3$ to the power of $5$ .

$\frac{2}{5} \cdot 243 - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.4

Simplify.

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Step 8.4.1

Combine $\frac{2}{5}$ and $243$ .

$\frac{2 \cdot 243}{5} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.4.2

Multiply $2$ by $243$ .

$\frac{486}{5} - \frac{2}{5} \cdot 4^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.5

Simplify.

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Step 8.5.1

Rewrite $4$ as $2^{2}$ .

$\frac{486}{5} - \frac{2}{5} \cdot {(2^{2})}^{\frac{5}{2}} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.5.2

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

$\frac{486}{5} - \frac{2}{5} \cdot 2^{2 (\frac{5}{2})} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.5.3

Cancel the common factor of $2$ .

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Step 8.5.3.1

Cancel the common factor.

$\frac{486}{5} - \frac{2}{5} \cdot 2^{2 (\frac{5}{2})} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.5.3.2

Rewrite the expression.

$\frac{486}{5} - \frac{2}{5} \cdot 2^{5} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.5.4

Raise $2$ to the power of $5$ .

$\frac{486}{5} - \frac{2}{5} \cdot 32 - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.6

Simplify.

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Step 8.6.1

Multiply $32$ by $- 1$ .

$\frac{486}{5} - 32 (\frac{2}{5}) - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.6.2

Combine $- 32$ and $\frac{2}{5}$ .

$\frac{486}{5} + \frac{- 32 \cdot 2}{5} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.6.3

Multiply $- 32$ by $2$ .

$\frac{486}{5} + \frac{- 64}{5} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.6.4

Move the negative in front of the fraction.

$\frac{486}{5} - \frac{64}{5} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.7

Simplify.

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Step 8.7.1

Combine the numerators over the common denominator.

$\frac{486 - 64}{5} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.7.2

Subtract $64$ from $486$ .

$\frac{422}{5} - 2 (\frac{2}{3} \cdot 9^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.8

Simplify.

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Step 8.8.1

Rewrite $9$ as $3^{2}$ .

$\frac{422}{5} - 2 (\frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.8.2

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

$\frac{422}{5} - 2 (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.8.3

Cancel the common factor of $2$ .

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Step 8.8.3.1

Cancel the common factor.

$\frac{422}{5} - 2 (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.8.3.2

Rewrite the expression.

$\frac{422}{5} - 2 (\frac{2}{3} \cdot 3^{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.8.4

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

$\frac{422}{5} - 2 (\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9

Simplify.

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Step 8.9.1

Combine $\frac{2}{3}$ and $27$ .

$\frac{422}{5} - 2 (\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9.2

Multiply $2$ by $27$ .

$\frac{422}{5} - 2 (\frac{54}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9.3

Cancel the common factor of $54$ and $3$ .

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Step 8.9.3.1

Factor $3$ out of $54$ .

$\frac{422}{5} - 2 (\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9.3.2

Cancel the common factors.

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Step 8.9.3.2.1

Factor $3$ out of $3$ .

$\frac{422}{5} - 2 (\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9.3.2.2

Cancel the common factor.

$\frac{422}{5} - 2 (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9.3.2.3

Rewrite the expression.

$\frac{422}{5} - 2 (\frac{18}{1} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.9.3.2.4

Divide $18$ by $1$ .

$\frac{422}{5} - 2 (18 - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 8.10

Simplify.

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Step 8.10.1

Rewrite $4$ as $2^{2}$ .

$\frac{422}{5} - 2 (18 - \frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}})$

Step 8.10.2

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

$\frac{422}{5} - 2 (18 - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})})$

Step 8.10.3

Cancel the common factor of $2$ .

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Step 8.10.3.1

Cancel the common factor.

$\frac{422}{5} - 2 (18 - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})})$

Step 8.10.3.2

Rewrite the expression.

$\frac{422}{5} - 2 (18 - \frac{2}{3} \cdot 2^{3})$

Step 8.10.4

Raise $2$ to the power of $3$ .

$\frac{422}{5} - 2 (18 - \frac{2}{3} \cdot 8)$

Step 8.11

Simplify.

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Step 8.11.1

Multiply $8$ by $- 1$ .

$\frac{422}{5} - 2 (18 - 8 (\frac{2}{3}))$

Step 8.11.2

Combine $- 8$ and $\frac{2}{3}$ .

$\frac{422}{5} - 2 (18 + \frac{- 8 \cdot 2}{3})$

Step 8.11.3

Multiply $- 8$ by $2$ .

$\frac{422}{5} - 2 (18 + \frac{- 16}{3})$

Step 8.11.4

Move the negative in front of the fraction.

$\frac{422}{5} - 2 (18 - \frac{16}{3})$

Step 8.12

Simplify.

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Step 8.12.1

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{422}{5} - 2 (18 \cdot \frac{3}{3} - \frac{16}{3})$

Step 8.12.2

Combine $18$ and $\frac{3}{3}$ .

$\frac{422}{5} - 2 (\frac{18 \cdot 3}{3} - \frac{16}{3})$

Step 8.12.3

Combine the numerators over the common denominator.

$\frac{422}{5} - 2 \frac{18 \cdot 3 - 16}{3}$

Step 8.12.4

Simplify the numerator.

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Step 8.12.4.1

Multiply $18$ by $3$ .

$\frac{422}{5} - 2 \frac{54 - 16}{3}$

Step 8.12.4.2

Subtract $16$ from $54$ .

$\frac{422}{5} - 2 (\frac{38}{3})$

Step 8.13

Simplify.

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Step 8.13.1

Combine $- 2$ and $\frac{38}{3}$ .

$\frac{422}{5} + \frac{- 2 \cdot 38}{3}$

Step 8.13.2

Multiply $- 2$ by $38$ .

$\frac{422}{5} + \frac{- 76}{3}$

Step 8.13.3

Move the negative in front of the fraction.

$\frac{422}{5} - \frac{76}{3}$

Step 8.14

Simplify.

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Step 8.14.1

To write $\frac{422}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{422}{5} \cdot \frac{3}{3} - \frac{76}{3}$

Step 8.14.2

To write $- \frac{76}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{422}{5} \cdot \frac{3}{3} - \frac{76}{3} \cdot \frac{5}{5}$

Step 8.14.3

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

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Step 8.14.3.1

Multiply $\frac{422}{5}$ by $\frac{3}{3}$ .

$\frac{422 \cdot 3}{5 \cdot 3} - \frac{76}{3} \cdot \frac{5}{5}$

Step 8.14.3.2

Multiply $5$ by $3$ .

$\frac{422 \cdot 3}{15} - \frac{76}{3} \cdot \frac{5}{5}$

Step 8.14.3.3

Multiply $\frac{76}{3}$ by $\frac{5}{5}$ .

$\frac{422 \cdot 3}{15} - \frac{76 \cdot 5}{3 \cdot 5}$

Step 8.14.3.4

Multiply $3$ by $5$ .

$\frac{422 \cdot 3}{15} - \frac{76 \cdot 5}{15}$

Step 8.14.4

Combine the numerators over the common denominator.

$\frac{422 \cdot 3 - 76 \cdot 5}{15}$

Step 8.14.5

Simplify the numerator.

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Step 8.14.5.1

Multiply $422$ by $3$ .

$\frac{1266 - 76 \cdot 5}{15}$

Step 8.14.5.2

Multiply $- 76$ by $5$ .

$\frac{1266 - 380}{15}$

Step 8.14.5.3

Subtract $380$ from $1266$ .

$\frac{886}{15}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{886}{15}$

Decimal Form:

$59.0\overline{6}$

Mixed Number Form:

$59 \frac{1}{15}$