Calculus Examples

$\int_{2}^{3} \frac{1}{\sqrt{x}} - 7 d x$

Step 1

Split the single integral into multiple integrals.

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

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

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

Step 2.2

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

$\int_{2}^{3} {(x^{\frac{1}{2}})}^{- 1} d x + \int_{2}^{3} - 7 d x$

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

$2 x^{\frac{1}{2}}]_{2}^{3} + \int_{2}^{3} - 7 d x$

Step 4

Apply the constant rule.

$2 x^{\frac{1}{2}}]_{2}^{3} + - 7 x]_{2}^{3}$

Step 5

Simplify the answer.

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Step 5.1

Combine $2 x^{\frac{1}{2}}]_{2}^{3}$ and $- 7 x]_{2}^{3}$ .

$2 x^{\frac{1}{2}} - 7 x]_{2}^{3}$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $2 x^{\frac{1}{2}} - 7 x$ at $3$ and at $2$ .

$(2 \cdot 3^{\frac{1}{2}} - 7 \cdot 3) - (2 \cdot 2^{\frac{1}{2}} - 7 \cdot 2)$

Step 5.2.2

Simplify.

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Step 5.2.2.1

Multiply $- 7$ by $3$ .

$2 \cdot 3^{\frac{1}{2}} - 21 - (2 \cdot 2^{\frac{1}{2}} - 7 \cdot 2)$

Step 5.2.2.2

Multiply $2$ by $2^{\frac{1}{2}}$ by adding the exponents.

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Step 5.2.2.2.1

Multiply $2$ by $2^{\frac{1}{2}}$ .

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Step 5.2.2.2.1.1

Raise $2$ to the power of $1$ .

$2 \cdot 3^{\frac{1}{2}} - 21 - (2^{1} \cdot 2^{\frac{1}{2}} - 7 \cdot 2)$

Step 5.2.2.2.1.2

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

$2 \cdot 3^{\frac{1}{2}} - 21 - (2^{1 + \frac{1}{2}} - 7 \cdot 2)$

Step 5.2.2.2.2

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

$2 \cdot 3^{\frac{1}{2}} - 21 - (2^{\frac{2}{2} + \frac{1}{2}} - 7 \cdot 2)$

Step 5.2.2.2.3

Combine the numerators over the common denominator.

$2 \cdot 3^{\frac{1}{2}} - 21 - (2^{\frac{2 + 1}{2}} - 7 \cdot 2)$

Step 5.2.2.2.4

Add $2$ and $1$ .

$2 \cdot 3^{\frac{1}{2}} - 21 - (2^{\frac{3}{2}} - 7 \cdot 2)$

Step 5.2.2.3

Multiply $- 7$ by $2$ .

$2 \cdot 3^{\frac{1}{2}} - 21 - (2^{\frac{3}{2}} - 14)$

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.

$2 \cdot 3^{\frac{1}{2}} - 21 - 2^{\frac{3}{2}} - - 14$

Step 5.3.1.2

Multiply $- 1$ by $- 14$ .

$2 \cdot 3^{\frac{1}{2}} - 21 - 2^{\frac{3}{2}} + 14$

Step 5.3.2

Add $- 21$ and $14$ .

$2 \cdot 3^{\frac{1}{2}} - 7 - 2^{\frac{3}{2}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 \cdot 3^{\frac{1}{2}} - 7 - 2^{\frac{3}{2}}$

Decimal Form:

$- 6.36432550 \dots$

Step 7