Calculus Examples

$\frac{d x}{9 x^{2} - 4} \cdot (d x)$

Step 1

Simplify.

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Step 1.1

Combine $d$ and $\frac{d x}{9 x^{2} - 4}$ .

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

Step 1.2

Raise $d$ to the power of $1$ .

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

Step 1.3

Raise $d$ to the power of $1$ .

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

Step 1.4

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

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

Step 1.5

Add $1$ and $1$ .

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

Step 1.6

Combine $\frac{d^{2} x}{9 x^{2} - 4}$ and $x$ .

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

Step 1.7

Raise $x$ to the power of $1$ .

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

Step 1.8

Raise $x$ to the power of $1$ .

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

Step 1.9

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

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

Step 1.10

Add $1$ and $1$ .

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

Step 2

Since $\frac{x^{2}}{9 x^{2} - 4}$ is constant with respect to $d$ , move $\frac{x^{2}}{9 x^{2} - 4}$ out of the integral.

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

Step 3

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

$\frac{x^{2}}{9 x^{2} - 4} (\frac{1}{3} d^{3} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{x^{2}}{9 x^{2} - 4} (\frac{1}{3} d^{3} + C)$ as $\frac{x^{2}}{(3 x + 2) (3 x - 2)} (\frac{1}{3} d^{3}) + C$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Combine $\frac{1}{3}$ and $d^{3}$ .

$\frac{x^{2}}{(3 x + 2) (3 x - 2)} \cdot \frac{d^{3}}{3} + C$

Step 4.2.2

Multiply $\frac{x^{2}}{(3 x + 2) (3 x - 2)}$ by $\frac{d^{3}}{3}$ .

$\frac{x^{2} d^{3}}{(3 x + 2) (3 x - 2) \cdot 3} + C$

Step 4.2.3

Move $3$ to the left of $(3 x + 2) (3 x - 2)$ .

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