Calculus Examples

$\int_{7}^{11} \frac{1}{\sqrt{x}} - 10 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{7}^{11} \frac{1}{\sqrt{x}} d x + \int_{7}^{11} - 10 d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\int_{7}^{11} \frac{1}{x^{\frac{1}{2}}} d x + \int_{7}^{11} - 10 d x$

Step 2.2

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

$\int_{7}^{11} {(x^{\frac{1}{2}})}^{- 1} d x + \int_{7}^{11} - 10 d x$

Step 2.3

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

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Step 2.3.1

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

$\int_{7}^{11} x^{\frac{1}{2} \cdot - 1} d x + \int_{7}^{11} - 10 d x$

Step 2.3.2

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

$\int_{7}^{11} x^{\frac{- 1}{2}} d x + \int_{7}^{11} - 10 d x$

Step 2.3.3

Move the negative in front of the fraction.

$\int_{7}^{11} x^{- \frac{1}{2}} d x + \int_{7}^{11} - 10 d x$

Step 3

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

$2 x^{\frac{1}{2}}]_{7}^{11} + \int_{7}^{11} - 10 d x$

Step 4

Apply the constant rule.

$2 x^{\frac{1}{2}}]_{7}^{11} + - 10 x]_{7}^{11}$

Step 5

Simplify the answer.

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Step 5.1

Combine $2 x^{\frac{1}{2}}]_{7}^{11}$ and $- 10 x]_{7}^{11}$ .

$2 x^{\frac{1}{2}} - 10 x]_{7}^{11}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $2 x^{\frac{1}{2}} - 10 x$ at $11$ and at $7$ .

$(2 \cdot 11^{\frac{1}{2}} - 10 \cdot 11) - (2 \cdot 7^{\frac{1}{2}} - 10 \cdot 7)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 10$ by $11$ .

$2 \cdot 11^{\frac{1}{2}} - 110 - (2 \cdot 7^{\frac{1}{2}} - 10 \cdot 7)$

Step 5.2.2.2

Multiply $- 10$ by $7$ .

$2 \cdot 11^{\frac{1}{2}} - 110 - (2 \cdot 7^{\frac{1}{2}} - 70)$

Step 5.3

Simplify.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Apply the distributive property.

$2 \cdot 11^{\frac{1}{2}} - 110 - (2 \cdot 7^{\frac{1}{2}}) - - 70$

Step 5.3.1.2

Multiply $2$ by $- 1$ .

$2 \cdot 11^{\frac{1}{2}} - 110 - 2 \cdot 7^{\frac{1}{2}} - - 70$

Step 5.3.1.3

Multiply $- 1$ by $- 70$ .

$2 \cdot 11^{\frac{1}{2}} - 110 - 2 \cdot 7^{\frac{1}{2}} + 70$

Step 5.3.2

Add $- 110$ and $70$ .

$2 \cdot 11^{\frac{1}{2}} - 40 - 2 \cdot 7^{\frac{1}{2}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \cdot 11^{\frac{1}{2}} - 40 - 2 \cdot 7^{\frac{1}{2}}$

Decimal Form:

$- 38.65825304 \dots$

Step 7