Calculus Examples

$\int \frac{7}{2 x^{\frac{9}{4}}} d x$

Step 1

Since $\frac{7}{2}$ is constant with respect to $x$ , move $\frac{7}{2}$ out of the integral.

$\frac{7}{2} \int \frac{1}{x^{\frac{9}{4}}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{\frac{9}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{7}{2} \int {(x^{\frac{9}{4}})}^{- 1} d x$

Step 2.2

Multiply the exponents in ${(x^{\frac{9}{4}})}^{- 1}$ .

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Step 2.2.1

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

$\frac{7}{2} \int x^{\frac{9}{4} \cdot - 1} d x$

Step 2.2.2

Multiply $\frac{9}{4} \cdot - 1$ .

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Step 2.2.2.1

Combine $\frac{9}{4}$ and $- 1$ .

$\frac{7}{2} \int x^{\frac{9 \cdot - 1}{4}} d x$

Step 2.2.2.2

Multiply $9$ by $- 1$ .

$\frac{7}{2} \int x^{\frac{- 9}{4}} d x$

Step 2.2.3

Move the negative in front of the fraction.

$\frac{7}{2} \int x^{- \frac{9}{4}} d x$

Step 3

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

$\frac{7}{2} (- \frac{4}{5} x^{- \frac{5}{4}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{7}{2} (- \frac{4}{5} x^{- \frac{5}{4}} + C)$ as $\frac{7}{2} (- \frac{4}{5} \cdot \frac{1}{x^{\frac{5}{4}}}) + C$ .

$\frac{7}{2} (- \frac{4}{5} \cdot \frac{1}{x^{\frac{5}{4}}}) + C$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{7}{2} (- \frac{4}{x^{\frac{5}{4}} \cdot 5}) + C$

Step 4.2.2

Move $5$ to the left of $x^{\frac{5}{4}}$ .

$\frac{7}{2} (- \frac{4}{5 \cdot x^{\frac{5}{4}}}) + C$

Step 4.2.3

Multiply $\frac{7}{2}$ by $\frac{4}{5 x^{\frac{5}{4}}}$ .

$- \frac{7 \cdot 4}{2 (5 x^{\frac{5}{4}})} + C$

Step 4.2.4

Multiply $7$ by $4$ .

$- \frac{28}{2 (5 x^{\frac{5}{4}})} + C$

Step 4.2.5

Multiply $5$ by $2$ .

$- \frac{28}{10 x^{\frac{5}{4}}} + C$

Step 4.2.6

Factor $2$ out of $28$ .

$- \frac{2 (14)}{10 x^{\frac{5}{4}}} + C$

Step 4.2.7

Cancel the common factors.

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Step 4.2.7.1

Factor $2$ out of $10 x^{\frac{5}{4}}$ .

$- \frac{2 (14)}{2 (5 x^{\frac{5}{4}})} + C$

Step 4.2.7.2

Cancel the common factor.

$- \frac{2 \cdot 14}{2 (5 x^{\frac{5}{4}})} + C$

Step 4.2.7.3

Rewrite the expression.

$- \frac{14}{5 x^{\frac{5}{4}}} + C$