Calculus Examples

$\int_{1}^{7} \frac{10 x^{2} + 9}{\sqrt{x}} d x$

Step 1

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

$\int_{1}^{7} \frac{10 x^{2} + 9}{x^{\frac{1}{2}}} d x$

Step 2

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

$\int_{1}^{7} (10 x^{2} + 9) {(x^{\frac{1}{2}})}^{- 1} d x$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{1}^{7} (10 x^{2} + 9) x^{\frac{1}{2} \cdot - 1} d x$

Step 3.2

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

$\int_{1}^{7} (10 x^{2} + 9) x^{\frac{- 1}{2}} d x$

Step 3.3

Move the negative in front of the fraction.

$\int_{1}^{7} (10 x^{2} + 9) x^{- \frac{1}{2}} d x$

Step 4

Expand $(10 x^{2} + 9) x^{- \frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

$\int_{1}^{7} 10 x^{2} x^{- \frac{1}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 4.2

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

$\int_{1}^{7} 10 x^{2 - \frac{1}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 4.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int_{1}^{7} 10 x^{2 \cdot \frac{2}{2} - \frac{1}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 4.4

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

$\int_{1}^{7} 10 x^{\frac{2 \cdot 2}{2} - \frac{1}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 4.5

Combine the numerators over the common denominator.

$\int_{1}^{7} 10 x^{\frac{2 \cdot 2 - 1}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $2$ .

$\int_{1}^{7} 10 x^{\frac{4 - 1}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 4.6.2

Subtract $1$ from $4$ .

$\int_{1}^{7} 10 x^{\frac{3}{2}} + 9 x^{- \frac{1}{2}} d x$

Step 5

Split the single integral into multiple integrals.

$\int_{1}^{7} 10 x^{\frac{3}{2}} d x + \int_{1}^{7} 9 x^{- \frac{1}{2}} d x$

Step 6

Since $10$ is constant with respect to $x$ , move $10$ out of the integral.

$10 \int_{1}^{7} x^{\frac{3}{2}} d x + \int_{1}^{7} 9 x^{- \frac{1}{2}} d x$

Step 7

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

$10 (\frac{2}{5} x^{\frac{5}{2}}]_{1}^{7}) + \int_{1}^{7} 9 x^{- \frac{1}{2}} d x$

Step 8

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$10 (\frac{2 x^{\frac{5}{2}}}{5}]_{1}^{7}) + \int_{1}^{7} 9 x^{- \frac{1}{2}} d x$

Step 9

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$10 (\frac{2 x^{\frac{5}{2}}}{5}]_{1}^{7}) + 9 \int_{1}^{7} x^{- \frac{1}{2}} d x$

Step 10

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

$10 (\frac{2 x^{\frac{5}{2}}}{5}]_{1}^{7}) + 9 (2 x^{\frac{1}{2}}]_{1}^{7})$

Step 11

Simplify the answer.

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Step 11.1

Substitute and simplify.

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Step 11.1.1

Evaluate $\frac{2 x^{\frac{5}{2}}}{5}$ at $7$ and at $1$ .

$10 ((\frac{2 \cdot 7^{\frac{5}{2}}}{5}) - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 9 (2 x^{\frac{1}{2}}]_{1}^{7})$

Step 11.1.2

Evaluate $2 x^{\frac{1}{2}}$ at $7$ and at $1$ .

$10 (\frac{2 \cdot 7^{\frac{5}{2}}}{5} - \frac{2 \cdot 1^{\frac{5}{2}}}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 11.1.3

Simplify.

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Step 11.1.3.1

One to any power is one.

$10 (\frac{2 \cdot 7^{\frac{5}{2}}}{5} - \frac{2 \cdot 1}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 11.1.3.2

Multiply $2$ by $1$ .

$10 (\frac{2 \cdot 7^{\frac{5}{2}}}{5} - \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 11.1.3.3

One to any power is one.

$10 (\frac{2 \cdot 7^{\frac{5}{2}}}{5} - \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2 \cdot 1)$

Step 11.1.3.4

Multiply $- 2$ by $1$ .

$10 (\frac{2 \cdot 7^{\frac{5}{2}}}{5} - \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2

Simplify.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Apply the distributive property.

$10 \frac{2 \cdot 7^{\frac{5}{2}}}{5} + 10 (- \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.2

Cancel the common factor of $5$ .

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Step 11.2.1.2.1

Factor $5$ out of $10$ .

$5 (2) \frac{2 \cdot 7^{\frac{5}{2}}}{5} + 10 (- \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.2.2

Cancel the common factor.

$5 \cdot 2 \frac{2 \cdot 7^{\frac{5}{2}}}{5} + 10 (- \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.2.3

Rewrite the expression.

$2 (2 \cdot 7^{\frac{5}{2}}) + 10 (- \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.3

Multiply $2$ by $2$ .

$4 \cdot 7^{\frac{5}{2}} + 10 (- \frac{2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.4

Cancel the common factor of $5$ .

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Step 11.2.1.4.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$4 \cdot 7^{\frac{5}{2}} + 10 (\frac{- 2}{5}) + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.4.2

Factor $5$ out of $10$ .

$4 \cdot 7^{\frac{5}{2}} + 5 (2) \frac{- 2}{5} + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.4.3

Cancel the common factor.

$4 \cdot 7^{\frac{5}{2}} + 5 \cdot 2 \frac{- 2}{5} + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.4.4

Rewrite the expression.

$4 \cdot 7^{\frac{5}{2}} + 2 \cdot - 2 + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.5

Multiply $2$ by $- 2$ .

$4 \cdot 7^{\frac{5}{2}} - 4 + 9 (2 \cdot 7^{\frac{1}{2}} - 2)$

Step 11.2.1.6

Apply the distributive property.

$4 \cdot 7^{\frac{5}{2}} - 4 + 9 (2 \cdot 7^{\frac{1}{2}}) + 9 \cdot - 2$

Step 11.2.1.7

Multiply $2$ by $9$ .

$4 \cdot 7^{\frac{5}{2}} - 4 + 18 \cdot 7^{\frac{1}{2}} + 9 \cdot - 2$

Step 11.2.1.8

Multiply $9$ by $- 2$ .

$4 \cdot 7^{\frac{5}{2}} - 4 + 18 \cdot 7^{\frac{1}{2}} - 18$

Step 11.2.2

Subtract $18$ from $- 4$ .

$4 \cdot 7^{\frac{5}{2}} + 18 \cdot 7^{\frac{1}{2}} - 22$

Step 12

The result can be shown in multiple forms.

Exact Form:

$4 \cdot 7^{\frac{5}{2}} + 18 \cdot 7^{\frac{1}{2}} - 22$

Decimal Form:

$544.19078056 \dots$

Step 13