Calculus Examples

$\int \frac{x^{3}}{5 - \frac{1}{2} x^{4}} d x$

Step 1

Let $u = 5 - \frac{1}{2} x^{4}$ . Then $d u = - 2 x^{3} d x$ , so $- \frac{1}{2} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 5 - \frac{1}{2} x^{4}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $5 - \frac{1}{2} x^{4}$ .

$\frac{d}{d x} [5 - \frac{1}{2} x^{4}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $5 - \frac{1}{2} x^{4}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- \frac{1}{2} x^{4}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- \frac{1}{2} x^{4}]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [- \frac{1}{2} x^{4}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{4}]$ .

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Step 1.1.3.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{4}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{4}]$ .

$0 - \frac{1}{2} \frac{d}{d x} [x^{4}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$0 - \frac{1}{2} (4 x^{3})$

Step 1.1.3.3

Multiply $4$ by $- 1$ .

$0 - 4 (\frac{1}{2}) x^{3}$

Step 1.1.3.4

Combine $- 4$ and $\frac{1}{2}$ .

$0 + \frac{- 4}{2} x^{3}$

Step 1.1.3.5

Combine $\frac{- 4}{2}$ and $x^{3}$ .

$0 + \frac{- 4 x^{3}}{2}$

Step 1.1.3.6

Cancel the common factor of $- 4$ and $2$ .

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Step 1.1.3.6.1

Factor $2$ out of $- 4 x^{3}$ .

$0 + \frac{2 (- 2 x^{3})}{2}$

Step 1.1.3.6.2

Cancel the common factors.

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Step 1.1.3.6.2.1

Factor $2$ out of $2$ .

$0 + \frac{2 (- 2 x^{3})}{2 (1)}$

Step 1.1.3.6.2.2

Cancel the common factor.

$0 + \frac{2 (- 2 x^{3})}{2 \cdot 1}$

Step 1.1.3.6.2.3

Rewrite the expression.

$0 + \frac{- 2 x^{3}}{1}$

Step 1.1.3.6.2.4

Divide $- 2 x^{3}$ by $1$ .

$0 - 2 x^{3}$

Step 1.1.4

Subtract $2 x^{3}$ from $0$ .

$- 2 x^{3}$

Step 1.2

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

$\int \frac{1}{u} \cdot \frac{1}{- 2} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int \frac{1}{u} (- \frac{1}{2}) d u$

Step 2.2

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$\int - \frac{1}{u \cdot 2} d u$

Step 2.3

Move $2$ to the left of $u$ .

$\int - \frac{1}{2 u} d u$

Step 3

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

$- \int \frac{1}{2 u} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int \frac{1}{u} d u)$

Step 5

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$- \frac{1}{2} (\ln (| u |) + C)$

Step 6

Simplify.

$- \frac{1}{2} \ln (| u |) + C$

Step 7

Replace all occurrences of $u$ with $5 - \frac{1}{2} x^{4}$ .

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