Calculus Examples

$\frac{d x}{{(3 - 2 x)}^{5}}$

Step 1

Write $\frac{d x}{{(3 - 2 x)}^{5}}$ as a function.

$f (d) = \frac{d x}{{(3 - 2 x)}^{5}}$

Step 2

The function $F (d)$ can be found by finding the indefinite integral of the derivative $f (d)$ .

$F (d) = \int f (d) d \cdot d$

Step 3

Set up the integral to solve.

$F (d) = \int \frac{d x}{{(3 - 2 x)}^{5}} d \cdot d$

Step 4

Since $\frac{x}{{(3 - 2 x)}^{5}}$ is constant with respect to $d$ , move $\frac{x}{{(3 - 2 x)}^{5}}$ out of the integral.

$\frac{x}{{(3 - 2 x)}^{5}} \int d d \cdot d$

Step 5

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

$\frac{x}{{(3 - 2 x)}^{5}} (\frac{1}{2} d^{2} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $\frac{x}{{(3 - 2 x)}^{5}} (\frac{1}{2} d^{2} + C)$ as $\frac{x}{{(3 - 2 x)}^{5}} \cdot \frac{1}{2} d^{2} + C$ .

$\frac{x}{{(3 - 2 x)}^{5}} \cdot \frac{1}{2} d^{2} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{x}{{(3 - 2 x)}^{5}}$ by $\frac{1}{2}$ .

$\frac{x}{{(3 - 2 x)}^{5} \cdot 2} d^{2} + C$

Step 6.2.2

Move $2$ to the left of ${(3 - 2 x)}^{5}$ .

$\frac{x}{2 \cdot {(3 - 2 x)}^{5}} d^{2} + C$

Step 6.2.3

Combine $\frac{x}{2 {(3 - 2 x)}^{5}}$ and $d^{2}$ .

$\frac{x d^{2}}{2 {(3 - 2 x)}^{5}} + C$

Step 7

The answer is the antiderivative of the function $f (d) = \frac{d x}{{(3 - 2 x)}^{5}}$ .

$F (d) =$ $\frac{x d^{2}}{2 {(3 - 2 x)}^{5}} + C$