Calculus Examples

$\int \frac{21 x^{2}}{4 {(x^{3} + 5)}^{\frac{5}{4}}} d x$

Step 1

Let $u = x^{3} + 5$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3 x^{2}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} + 5$ .

$\frac{d}{d x} [x^{3} + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [5]$

Step 1.1.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

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

$\int \frac{7}{4 u^{\frac{5}{4}}} d u$

Step 2

Since $\frac{7}{4}$ is constant with respect to $u$ , move $\frac{7}{4}$ out of the integral.

$\frac{7}{4} \int \frac{1}{u^{\frac{5}{4}}} d u$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $u^{\frac{5}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{7}{4} \int {(u^{\frac{5}{4}})}^{- 1} d u$

Step 3.2

Multiply the exponents in ${(u^{\frac{5}{4}})}^{- 1}$ .

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Step 3.2.1

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

$\frac{7}{4} \int u^{\frac{5}{4} \cdot - 1} d u$

Step 3.2.2

Multiply $\frac{5}{4} \cdot - 1$ .

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Step 3.2.2.1

Combine $\frac{5}{4}$ and $- 1$ .

$\frac{7}{4} \int u^{\frac{5 \cdot - 1}{4}} d u$

Step 3.2.2.2

Multiply $5$ by $- 1$ .

$\frac{7}{4} \int u^{\frac{- 5}{4}} d u$

Step 3.2.3

Move the negative in front of the fraction.

$\frac{7}{4} \int u^{- \frac{5}{4}} d u$

Step 4

By the Power Rule, the integral of $u^{- \frac{5}{4}}$ with respect to $u$ is $- 4 u^{- \frac{1}{4}}$ .

$\frac{7}{4} (- 4 u^{- \frac{1}{4}} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{7}{4} (- 4 u^{- \frac{1}{4}} + C)$ as $\frac{7}{4} (- 4 \frac{1}{u^{\frac{1}{4}}}) + C$ .

$\frac{7}{4} (- 4 \frac{1}{u^{\frac{1}{4}}}) + C$

Step 5.2

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

$\frac{7}{4} \cdot \frac{- 4}{u^{\frac{1}{4}}} + C$

Step 5.3

Simplify.

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Step 5.3.1

Move the negative in front of the fraction.

$\frac{7}{4} (- \frac{4}{u^{\frac{1}{4}}}) + C$

Step 5.3.2

Multiply $\frac{7}{4}$ by $\frac{4}{u^{\frac{1}{4}}}$ .

$- \frac{7 \cdot 4}{4 u^{\frac{1}{4}}} + C$

Step 5.3.3

Multiply $7$ by $4$ .

$- \frac{28}{4 u^{\frac{1}{4}}} + C$

Step 5.3.4

Factor $4$ out of $28$ .

$- \frac{4 \cdot 7}{4 u^{\frac{1}{4}}} + C$

Step 5.3.5

Cancel the common factors.

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Step 5.3.5.1

Factor $4$ out of $4 u^{\frac{1}{4}}$ .

$- \frac{4 \cdot 7}{4 (u^{\frac{1}{4}})} + C$

Step 5.3.5.2

Cancel the common factor.

$- \frac{4 \cdot 7}{4 u^{\frac{1}{4}}} + C$

Step 5.3.5.3

Rewrite the expression.

$- \frac{7}{u^{\frac{1}{4}}} + C$

Step 6

Replace all occurrences of $u$ with $x^{3} + 5$ .

$- \frac{7}{{(x^{3} + 5)}^{\frac{1}{4}}} + C$