Calculus Examples

$\int_{0}^{1} \frac{52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}} + 22 x^{\frac{3}{2}}}{\sqrt{x}} d x$

Step 1

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

$\int_{0}^{1} \frac{52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}} + 22 x^{\frac{3}{2}}}{x^{\frac{1}{2}}} d x$

Step 2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{0}^{1} (52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}} + 22 x^{\frac{3}{2}}) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 3.1

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

$\int_{0}^{1} (52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}} + 22 x^{\frac{3}{2}}) x^{\frac{1}{2} \cdot - 1} d x$

Step 3.2

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

$\int_{0}^{1} (52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}} + 22 x^{\frac{3}{2}}) x^{\frac{- 1}{2}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int_{0}^{1} (52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}} + 22 x^{\frac{3}{2}}) x^{- \frac{1}{2}} d x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\int_{0}^{1} (52 x^{\frac{7}{2}} - 66 x^{\frac{5}{2}}) x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.2

Apply the distributive property.

$\int_{0}^{1} 52 x^{\frac{7}{2}} x^{- \frac{1}{2}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.3

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

$\int_{0}^{1} 52 x^{\frac{7}{2} - \frac{1}{2}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.4

Combine the numerators over the common denominator.

$\int_{0}^{1} 52 x^{\frac{7 - 1}{2}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.5

Subtract $1$ from $7$ .

$\int_{0}^{1} 52 x^{\frac{6}{2}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.6

Cancel the common factor of $6$ and $2$ .

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Step 4.6.1

Factor $2$ out of $6$ .

$\int_{0}^{1} 52 x^{\frac{2 \cdot 3}{2}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $2$ out of $2$ .

$\int_{0}^{1} 52 x^{\frac{2 \cdot 3}{2 (1)}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.6.2.2

Cancel the common factor.

$\int_{0}^{1} 52 x^{\frac{2 \cdot 3}{2 \cdot 1}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.6.2.3

Rewrite the expression.

$\int_{0}^{1} 52 x^{\frac{3}{1}} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.6.2.4

Divide $3$ by $1$ .

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{5}{2}} x^{- \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.7

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

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{5}{2} - \frac{1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.8

Combine the numerators over the common denominator.

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{5 - 1}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.9

Subtract $1$ from $5$ .

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{4}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.10

Cancel the common factor of $4$ and $2$ .

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Step 4.10.1

Factor $2$ out of $4$ .

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{2 \cdot 2}{2}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.10.2

Cancel the common factors.

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Step 4.10.2.1

Factor $2$ out of $2$ .

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{2 \cdot 2}{2 (1)}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.10.2.2

Cancel the common factor.

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{2 \cdot 2}{2 \cdot 1}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.10.2.3

Rewrite the expression.

$\int_{0}^{1} 52 x^{3} - 66 x^{\frac{2}{1}} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.10.2.4

Divide $2$ by $1$ .

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x^{\frac{3}{2}} x^{- \frac{1}{2}} d x$

Step 4.11

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

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x^{\frac{3}{2} - \frac{1}{2}} d x$

Step 4.12

Combine the numerators over the common denominator.

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x^{\frac{3 - 1}{2}} d x$

Step 4.13

Subtract $1$ from $3$ .

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x^{\frac{2}{2}} d x$

Step 4.14

Cancel the common factor of $2$ .

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Step 4.14.1

Cancel the common factor.

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x^{\frac{2}{2}} d x$

Step 4.14.2

Rewrite the expression.

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x^{1} d x$

Step 4.15

Simplify.

$\int_{0}^{1} 52 x^{3} - 66 x^{2} + 22 x d x$

Step 5

Split the single integral into multiple integrals.

$\int_{0}^{1} 52 x^{3} d x + \int_{0}^{1} - 66 x^{2} d x + \int_{0}^{1} 22 x d x$

Step 6

Since $52$ is constant with respect to $x$ , move $52$ out of the integral.

$52 \int_{0}^{1} x^{3} d x + \int_{0}^{1} - 66 x^{2} d x + \int_{0}^{1} 22 x d x$

Step 7

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$52 (\frac{1}{4} x^{4}]_{0}^{1}) + \int_{0}^{1} - 66 x^{2} d x + \int_{0}^{1} 22 x d x$

Step 8

Combine $\frac{1}{4}$ and $x^{4}$ .

$52 (\frac{x^{4}}{4}]_{0}^{1}) + \int_{0}^{1} - 66 x^{2} d x + \int_{0}^{1} 22 x d x$

Step 9

Since $- 66$ is constant with respect to $x$ , move $- 66$ out of the integral.

$52 (\frac{x^{4}}{4}]_{0}^{1}) - 66 \int_{0}^{1} x^{2} d x + \int_{0}^{1} 22 x d x$

Step 10

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

$52 (\frac{x^{4}}{4}]_{0}^{1}) - 66 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} 22 x d x$

Step 11

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

$52 (\frac{x^{4}}{4}]_{0}^{1}) - 66 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} 22 x d x$

Step 12

Since $22$ is constant with respect to $x$ , move $22$ out of the integral.

$52 (\frac{x^{4}}{4}]_{0}^{1}) - 66 (\frac{x^{3}}{3}]_{0}^{1}) + 22 \int_{0}^{1} x d x$

Step 13

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

$52 (\frac{x^{4}}{4}]_{0}^{1}) - 66 (\frac{x^{3}}{3}]_{0}^{1}) + 22 (\frac{1}{2} x^{2}]_{0}^{1})$

Step 14

Simplify the answer.

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Step 14.1

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

$52 (\frac{x^{4}}{4}]_{0}^{1}) - 66 (\frac{x^{3}}{3}]_{0}^{1}) + 22 (\frac{x^{2}}{2}]_{0}^{1})$

Step 14.2

Substitute and simplify.

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Step 14.2.1

Evaluate $\frac{x^{4}}{4}$ at $1$ and at $0$ .

$52 ((\frac{1^{4}}{4}) - \frac{0^{4}}{4}) - 66 (\frac{x^{3}}{3}]_{0}^{1}) + 22 (\frac{x^{2}}{2}]_{0}^{1})$

Step 14.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $0$ .

$52 (\frac{1^{4}}{4} - \frac{0^{4}}{4}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 (\frac{x^{2}}{2}]_{0}^{1})$

Step 14.2.3

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

$52 (\frac{1^{4}}{4} - \frac{0^{4}}{4}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4

Simplify.

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Step 14.2.4.1

One to any power is one.

$52 (\frac{1}{4} - \frac{0^{4}}{4}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.2

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

$52 (\frac{1}{4} - \frac{0}{4}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.3

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

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Step 14.2.4.3.1

Factor $4$ out of $0$ .

$52 (\frac{1}{4} - \frac{4 (0)}{4}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.3.2

Cancel the common factors.

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Step 14.2.4.3.2.1

Factor $4$ out of $4$ .

$52 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.3.2.2

Cancel the common factor.

$52 (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.3.2.3

Rewrite the expression.

$52 (\frac{1}{4} - \frac{0}{1}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.3.2.4

Divide $0$ by $1$ .

$52 (\frac{1}{4} - 0) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.4

Multiply $- 1$ by $0$ .

$52 (\frac{1}{4} + 0) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.5

Add $\frac{1}{4}$ and $0$ .

$52 (\frac{1}{4}) - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.6

Combine $52$ and $\frac{1}{4}$ .

$\frac{52}{4} - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.7

Cancel the common factor of $52$ and $4$ .

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Step 14.2.4.7.1

Factor $4$ out of $52$ .

$\frac{4 \cdot 13}{4} - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.7.2

Cancel the common factors.

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Step 14.2.4.7.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 13}{4 (1)} - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.7.2.2

Cancel the common factor.

$\frac{4 \cdot 13}{4 \cdot 1} - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.7.2.3

Rewrite the expression.

$\frac{13}{1} - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.7.2.4

Divide $13$ by $1$ .

$13 - 66 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.8

One to any power is one.

$13 - 66 (\frac{1}{3} - \frac{0^{3}}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.9

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

$13 - 66 (\frac{1}{3} - \frac{0}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.10

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

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Step 14.2.4.10.1

Factor $3$ out of $0$ .

$13 - 66 (\frac{1}{3} - \frac{3 (0)}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.10.2

Cancel the common factors.

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Step 14.2.4.10.2.1

Factor $3$ out of $3$ .

$13 - 66 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.10.2.2

Cancel the common factor.

$13 - 66 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.10.2.3

Rewrite the expression.

$13 - 66 (\frac{1}{3} - \frac{0}{1}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.10.2.4

Divide $0$ by $1$ .

$13 - 66 (\frac{1}{3} - 0) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.11

Multiply $- 1$ by $0$ .

$13 - 66 (\frac{1}{3} + 0) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.12

Add $\frac{1}{3}$ and $0$ .

$13 - 66 (\frac{1}{3}) + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.13

Combine $- 66$ and $\frac{1}{3}$ .

$13 + \frac{- 66}{3} + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.14

Cancel the common factor of $- 66$ and $3$ .

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Step 14.2.4.14.1

Factor $3$ out of $- 66$ .

$13 + \frac{3 \cdot - 22}{3} + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.14.2

Cancel the common factors.

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Step 14.2.4.14.2.1

Factor $3$ out of $3$ .

$13 + \frac{3 \cdot - 22}{3 (1)} + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.14.2.2

Cancel the common factor.

$13 + \frac{3 \cdot - 22}{3 \cdot 1} + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.14.2.3

Rewrite the expression.

$13 + \frac{- 22}{1} + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.14.2.4

Divide $- 22$ by $1$ .

$13 - 22 + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.15

Subtract $22$ from $13$ .

$- 9 + 22 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2})$

Step 14.2.4.16

One to any power is one.

$- 9 + 22 (\frac{1}{2} - \frac{0^{2}}{2})$

Step 14.2.4.17

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

$- 9 + 22 (\frac{1}{2} - \frac{0}{2})$

Step 14.2.4.18

Cancel the common factor of $0$ and $2$ .

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Step 14.2.4.18.1

Factor $2$ out of $0$ .

$- 9 + 22 (\frac{1}{2} - \frac{2 (0)}{2})$

Step 14.2.4.18.2

Cancel the common factors.

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Step 14.2.4.18.2.1

Factor $2$ out of $2$ .

$- 9 + 22 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 14.2.4.18.2.2

Cancel the common factor.

$- 9 + 22 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 14.2.4.18.2.3

Rewrite the expression.

$- 9 + 22 (\frac{1}{2} - \frac{0}{1})$

Step 14.2.4.18.2.4

Divide $0$ by $1$ .

$- 9 + 22 (\frac{1}{2} - 0)$

Step 14.2.4.19

Multiply $- 1$ by $0$ .

$- 9 + 22 (\frac{1}{2} + 0)$

Step 14.2.4.20

Add $\frac{1}{2}$ and $0$ .

$- 9 + 22 (\frac{1}{2})$

Step 14.2.4.21

Combine $22$ and $\frac{1}{2}$ .

$- 9 + \frac{22}{2}$

Step 14.2.4.22

Cancel the common factor of $22$ and $2$ .

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Step 14.2.4.22.1

Factor $2$ out of $22$ .

$- 9 + \frac{2 \cdot 11}{2}$

Step 14.2.4.22.2

Cancel the common factors.

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Step 14.2.4.22.2.1

Factor $2$ out of $2$ .

$- 9 + \frac{2 \cdot 11}{2 (1)}$

Step 14.2.4.22.2.2

Cancel the common factor.

$- 9 + \frac{2 \cdot 11}{2 \cdot 1}$

Step 14.2.4.22.2.3

Rewrite the expression.

$- 9 + \frac{11}{1}$

Step 14.2.4.22.2.4

Divide $11$ by $1$ .

$- 9 + 11$

Step 14.2.4.23

Add $- 9$ and $11$ .

$2$

Step 15