Calculus Examples

$\int_{1}^{32} x^{- \frac{9}{5}} d x$

Step 1

By the Power Rule, the integral of $x^{- \frac{9}{5}}$ with respect to $x$ is $- \frac{5}{4} x^{- \frac{4}{5}}$ .

$- \frac{5}{4} x^{- \frac{4}{5}}]_{1}^{32}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $- \frac{5}{4} x^{- \frac{4}{5}}$ at $32$ and at $1$ .

$(- \frac{5}{4} \cdot 32^{- \frac{4}{5}}) + \frac{5}{4} \cdot 1^{- \frac{4}{5}}$

Step 2.2

Simplify.

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Step 2.2.1

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

$- \frac{5}{4} \cdot \frac{1}{32^{\frac{4}{5}}} + \frac{5}{4} \cdot 1^{- \frac{4}{5}}$

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Cancel the common factor of $5$ .

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Raise $2$ to the power of $4$ .

$- \frac{5}{4} \cdot \frac{1}{16} + \frac{5}{4} \cdot 1^{- \frac{4}{5}}$

Step 2.2.6

Multiply $\frac{1}{16}$ by $\frac{5}{4}$ .

$- \frac{5}{16 \cdot 4} + \frac{5}{4} \cdot 1^{- \frac{4}{5}}$

Step 2.2.7

Multiply $16$ by $4$ .

$- \frac{5}{64} + \frac{5}{4} \cdot 1^{- \frac{4}{5}}$

Step 2.2.8

One to any power is one.

$- \frac{5}{64} + \frac{5}{4} \cdot 1$

Step 2.2.9

Multiply $\frac{5}{4}$ by $1$ .

$- \frac{5}{64} + \frac{5}{4}$

Step 2.2.10

To write $\frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$- \frac{5}{64} + \frac{5}{4} \cdot \frac{16}{16}$

Step 2.2.11

Write each expression with a common denominator of $64$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.11.1

Multiply $\frac{5}{4}$ by $\frac{16}{16}$ .

$- \frac{5}{64} + \frac{5 \cdot 16}{4 \cdot 16}$

Step 2.2.11.2

Multiply $4$ by $16$ .

$- \frac{5}{64} + \frac{5 \cdot 16}{64}$

Step 2.2.12

Combine the numerators over the common denominator.

$\frac{- 5 + 5 \cdot 16}{64}$

Step 2.2.13

Simplify the numerator.

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Step 2.2.13.1

Multiply $5$ by $16$ .

$\frac{- 5 + 80}{64}$

Step 2.2.13.2

Add $- 5$ and $80$ .

$\frac{75}{64}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{75}{64}$

Decimal Form:

$1.171875$

Mixed Number Form:

$1 \frac{11}{64}$

Step 4