Calculus Examples

$\int \frac{15}{{(3 x - 10)}^{\frac{4}{3}}} d x$

Step 1

Let $u = 3 x - 10$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x - 10$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Rewrite.

$\frac{3}{1}$

Step 1.1.2

Divide $3$ by $1$ .

$3$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{5}{u^{\frac{4}{3}}} d u$

Step 2

Since $5$ is constant with respect to $u$ , move $5$ out of the integral.

$5 \int \frac{1}{u^{\frac{4}{3}}} d u$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $u^{\frac{4}{3}}$ out of the denominator by raising it to the $- 1$ power.

$5 \int {(u^{\frac{4}{3}})}^{- 1} d u$

Step 3.2

Multiply the exponents in ${(u^{\frac{4}{3}})}^{- 1}$ .

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Step 3.2.1

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

$5 \int u^{\frac{4}{3} \cdot - 1} d u$

Step 3.2.2

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 3.2.2.1

Combine $\frac{4}{3}$ and $- 1$ .

$5 \int u^{\frac{4 \cdot - 1}{3}} d u$

Step 3.2.2.2

Multiply $4$ by $- 1$ .

$5 \int u^{\frac{- 4}{3}} d u$

Step 3.2.3

Move the negative in front of the fraction.

$5 \int u^{- \frac{4}{3}} d u$

Step 4

By the Power Rule, the integral of $u^{- \frac{4}{3}}$ with respect to $u$ is $- 3 u^{- \frac{1}{3}}$ .

$5 (- 3 u^{- \frac{1}{3}} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $5 (- 3 u^{- \frac{1}{3}} + C)$ as $5 (- 3 \frac{1}{u^{\frac{1}{3}}}) + C$ .

$5 (- 3 \frac{1}{u^{\frac{1}{3}}}) + C$

Step 5.2

Rewrite $5 (- 3 \frac{1}{u^{\frac{1}{3}}}) + C$ as $5 \frac{- 3}{u^{\frac{1}{3}}} + C$ .

$5 \frac{- 3}{u^{\frac{1}{3}}} + C$

Step 5.3

Simplify.

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Step 5.3.1

Move the negative in front of the fraction.

$5 (- \frac{3}{u^{\frac{1}{3}}}) + C$

Step 5.3.2

Multiply $- 1$ by $5$ .

$- 5 \frac{3}{u^{\frac{1}{3}}} + C$

Step 5.3.3

Combine $- 5$ and $\frac{3}{u^{\frac{1}{3}}}$ .

$\frac{- 5 \cdot 3}{u^{\frac{1}{3}}} + C$

Step 5.3.4

Multiply $- 5$ by $3$ .

$\frac{- 15}{u^{\frac{1}{3}}} + C$

Step 5.3.5

Move the negative in front of the fraction.

$- \frac{15}{u^{\frac{1}{3}}} + C$

Step 6

Replace all occurrences of $u$ with $3 x - 10$ .

$- \frac{15}{{(3 x - 10)}^{\frac{1}{3}}} + C$