Calculus Examples

$\int_{3}^{5} \frac{1}{\sqrt{x}} - 4 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{3}^{5} \frac{1}{\sqrt{x}} d x + \int_{3}^{5} - 4 d x$

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

$\int_{3}^{5} \frac{1}{x^{\frac{1}{2}}} d x + \int_{3}^{5} - 4 d x$

Step 2.2

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

$\int_{3}^{5} {(x^{\frac{1}{2}})}^{- 1} d x + \int_{3}^{5} - 4 d x$

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

$\int_{3}^{5} x^{\frac{1}{2} \cdot - 1} d x + \int_{3}^{5} - 4 d x$

Step 2.3.2

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

$\int_{3}^{5} x^{\frac{- 1}{2}} d x + \int_{3}^{5} - 4 d x$

Step 2.3.3

Move the negative in front of the fraction.

$\int_{3}^{5} x^{- \frac{1}{2}} d x + \int_{3}^{5} - 4 d x$

Step 3

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

$2 x^{\frac{1}{2}}]_{3}^{5} + \int_{3}^{5} - 4 d x$

Step 4

Apply the constant rule.

$2 x^{\frac{1}{2}}]_{3}^{5} + - 4 x]_{3}^{5}$

Step 5

Simplify the answer.

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Step 5.1

Combine $2 x^{\frac{1}{2}}]_{3}^{5}$ and $- 4 x]_{3}^{5}$ .

$2 x^{\frac{1}{2}} - 4 x]_{3}^{5}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $2 x^{\frac{1}{2}} - 4 x$ at $5$ and at $3$ .

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 4$ by $5$ .

$2 \cdot 5^{\frac{1}{2}} - 20 - (2 \cdot 3^{\frac{1}{2}} - 4 \cdot 3)$

Step 5.2.2.2

Multiply $- 4$ by $3$ .

$2 \cdot 5^{\frac{1}{2}} - 20 - (2 \cdot 3^{\frac{1}{2}} - 12)$

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.

$2 \cdot 5^{\frac{1}{2}} - 20 - (2 \cdot 3^{\frac{1}{2}}) - - 12$

Step 5.3.1.2

Multiply $2$ by $- 1$ .

$2 \cdot 5^{\frac{1}{2}} - 20 - 2 \cdot 3^{\frac{1}{2}} - - 12$

Step 5.3.1.3

Multiply $- 1$ by $- 12$ .

$2 \cdot 5^{\frac{1}{2}} - 20 - 2 \cdot 3^{\frac{1}{2}} + 12$

Step 5.3.2

Add $- 20$ and $12$ .

$2 \cdot 5^{\frac{1}{2}} - 8 - 2 \cdot 3^{\frac{1}{2}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \cdot 5^{\frac{1}{2}} - 8 - 2 \cdot 3^{\frac{1}{2}}$

Decimal Form:

$- 6.99196566 \dots$

Step 7