Calculus Examples

$\int_{1}^{32} x^{\frac{- 6}{5}} d x$

Step 1

Move the negative in front of the fraction.

$\int_{1}^{32} x^{- \frac{6}{5}} d x$

Step 2

By the Power Rule, the integral of $x^{- \frac{6}{5}}$ with respect to $x$ is $- 5 x^{- \frac{1}{5}}$ .

$- 5 x^{- \frac{1}{5}}]_{1}^{32}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $- 5 x^{- \frac{1}{5}}$ at $32$ and at $1$ .

$(- 5 \cdot 32^{- \frac{1}{5}}) + 5 \cdot 1^{- \frac{1}{5}}$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 \cdot \frac{1}{32^{\frac{1}{5}}} + 5 \cdot 1^{- \frac{1}{5}}$

Step 3.2.2

Rewrite $32$ as $2^{5}$ .

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

Step 3.2.3

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

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

Step 3.2.4

Cancel the common factor of $5$ .

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Step 3.2.4.1

Cancel the common factor.

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

Step 3.2.4.2

Rewrite the expression.

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

Step 3.2.5

Evaluate the exponent.

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

Step 3.2.6

Combine $- 5$ and $\frac{1}{2}$ .

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

Step 3.2.7

Move the negative in front of the fraction.

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

Step 3.2.8

One to any power is one.

$- \frac{5}{2} + 5 \cdot 1$

Step 3.2.9

Multiply $5$ by $1$ .

$- \frac{5}{2} + 5$

Step 3.2.10

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{5}{2} + 5 \cdot \frac{2}{2}$

Step 3.2.11

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

$- \frac{5}{2} + \frac{5 \cdot 2}{2}$

Step 3.2.12

Combine the numerators over the common denominator.

$\frac{- 5 + 5 \cdot 2}{2}$

Step 3.2.13

Simplify the numerator.

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Step 3.2.13.1

Multiply $5$ by $2$ .

$\frac{- 5 + 10}{2}$

Step 3.2.13.2

Add $- 5$ and $10$ .

$\frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{2}$

Decimal Form:

$2.5$

Mixed Number Form:

$2 \frac{1}{2}$

Step 5