Calculus Examples

$\int \frac{9 - 7 x^{\frac{3}{2}}}{- 7 x^{\frac{2}{3}}} d x$

Step 1

Move the negative in front of the fraction.

$\int - \frac{9 - 7 x^{\frac{3}{2}}}{7 x^{\frac{2}{3}}} d x$

Step 2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int \frac{9 - 7 x^{\frac{3}{2}}}{7 x^{\frac{2}{3}}} d x$

Step 3

Since $\frac{1}{7}$ is constant with respect to $x$ , move $\frac{1}{7}$ out of the integral.

$- (\frac{1}{7} \int \frac{9 - 7 x^{\frac{3}{2}}}{x^{\frac{2}{3}}} d x)$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{\frac{2}{3}}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{7} \int (9 - 7 x^{\frac{3}{2}}) {(x^{\frac{2}{3}})}^{- 1} d x$

Step 4.2

Multiply the exponents in ${(x^{\frac{2}{3}})}^{- 1}$ .

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Step 4.2.1

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

$- \frac{1}{7} \int (9 - 7 x^{\frac{3}{2}}) x^{\frac{2}{3} \cdot - 1} d x$

Step 4.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 4.2.2.1

Combine $\frac{2}{3}$ and $- 1$ .

$- \frac{1}{7} \int (9 - 7 x^{\frac{3}{2}}) x^{\frac{2 \cdot - 1}{3}} d x$

Step 4.2.2.2

Multiply $2$ by $- 1$ .

$- \frac{1}{7} \int (9 - 7 x^{\frac{3}{2}}) x^{\frac{- 2}{3}} d x$

Step 4.2.3

Move the negative in front of the fraction.

$- \frac{1}{7} \int (9 - 7 x^{\frac{3}{2}}) x^{- \frac{2}{3}} d x$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3}{2}} x^{- \frac{2}{3}} d x$

Step 5.2

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

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3}{2} - \frac{2}{3}} d x$

Step 5.3

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3}{2} \cdot \frac{3}{3} - \frac{2}{3}} d x$

Step 5.4

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3}{2} \cdot \frac{3}{3} - \frac{2}{3} \cdot \frac{2}{2}} d x$

Step 5.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.5.1

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3 \cdot 3}{2 \cdot 3} - \frac{2}{3} \cdot \frac{2}{2}} d x$

Step 5.5.2

Multiply $2$ by $3$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3 \cdot 3}{6} - \frac{2}{3} \cdot \frac{2}{2}} d x$

Step 5.5.3

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3 \cdot 3}{6} - \frac{2 \cdot 2}{3 \cdot 2}} d x$

Step 5.5.4

Multiply $3$ by $2$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3 \cdot 3}{6} - \frac{2 \cdot 2}{6}} d x$

Step 5.6

Combine the numerators over the common denominator.

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{3 \cdot 3 - 2 \cdot 2}{6}} d x$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Multiply $3$ by $3$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{9 - 2 \cdot 2}{6}} d x$

Step 5.7.2

Multiply $- 2$ by $2$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{9 - 4}{6}} d x$

Step 5.7.3

Subtract $4$ from $9$ .

$- \frac{1}{7} \int 9 x^{- \frac{2}{3}} - 7 x^{\frac{5}{6}} d x$

Step 6

Split the single integral into multiple integrals.

$- \frac{1}{7} (\int 9 x^{- \frac{2}{3}} d x + \int - 7 x^{\frac{5}{6}} d x)$

Step 7

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$- \frac{1}{7} (9 \int x^{- \frac{2}{3}} d x + \int - 7 x^{\frac{5}{6}} d x)$

Step 8

By the Power Rule, the integral of $x^{- \frac{2}{3}}$ with respect to $x$ is $3 x^{\frac{1}{3}}$ .

$- \frac{1}{7} (9 (3 x^{\frac{1}{3}} + C) + \int - 7 x^{\frac{5}{6}} d x)$

Step 9

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$- \frac{1}{7} (9 (3 x^{\frac{1}{3}} + C) - 7 \int x^{\frac{5}{6}} d x)$

Step 10

By the Power Rule, the integral of $x^{\frac{5}{6}}$ with respect to $x$ is $\frac{6}{11} x^{\frac{11}{6}}$ .

$- \frac{1}{7} (9 (3 x^{\frac{1}{3}} + C) - 7 (\frac{6}{11} x^{\frac{11}{6}} + C))$

Step 11

Simplify.

$- \frac{1}{7} (27 x^{\frac{1}{3}} - \frac{42 x^{\frac{11}{6}}}{11}) + C$

Step 12

Reorder terms.

$- \frac{1}{7} (27 x^{\frac{1}{3}} - \frac{42}{11} x^{\frac{11}{6}}) + C$