Calculus Examples

$\int \frac{1}{4^{2 a x}} d x$

Step 1

Simplify the expression.

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Step 1.1

Negate the exponent of $4^{2 a x}$ and move it out of the denominator.

$\int 1 {(4^{2 a x})}^{- 1} d x$

Step 1.2

Simplify.

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Step 1.2.1

Multiply the exponents in ${(4^{2 a x})}^{- 1}$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

Multiply $- 1$ by $2$ .

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

Step 1.2.2

Multiply $4^{- 2 a x}$ by $1$ .

$\int 4^{- 2 a x} d x$

Step 2

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

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Step 2.1

Let $u = - 2 a x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $- 2 a x$ .

$\frac{d}{d x} [- 2 a x]$

Step 2.1.2

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

$- 2 a \frac{d}{d x} [x]$

Step 2.1.3

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

$- 2 a \cdot 1$

Step 2.1.4

Multiply $- 2$ by $1$ .

$- 2 a$

Step 2.2

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

$\int 4^{u} \cdot \frac{1}{- 2 a} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$\int 4^{u} \cdot (- \frac{1}{2 a}) d u$

Step 3.2

Combine $4^{u}$ and $\frac{1}{2 a}$ .

$\int - \frac{4^{u}}{2 a} d u$

Step 4

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

$- \int \frac{4^{u}}{2 a} d u$

Step 5

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

$- (\frac{1}{2 a} \int 4^{u} d u)$

Step 6

The integral of $4^{u}$ with respect to $u$ is $\frac{4^{u}}{\ln (4)}$ .

$- \frac{1}{2 a} (\frac{4^{u}}{\ln (4)} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $- \frac{1}{2 a} (\frac{4^{u}}{\ln (4)} + C)$ as $- \frac{1}{2 a} \cdot \frac{1}{\ln (4)} \cdot 4^{u} + C$ .

$- \frac{1}{2 a} \cdot \frac{1}{\ln (4)} \cdot 4^{u} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{1}{\ln (4)}$ by $\frac{1}{2 a}$ .

$- \frac{1}{\ln (4) (2 a)} \cdot 4^{u} + C$

Step 7.2.2

Move $2$ to the left of $\ln (4)$ .

$- \frac{1}{2 \cdot \ln (4) a} \cdot 4^{u} + C$

Step 7.2.3

Combine $4^{u}$ and $\frac{1}{2 \ln (4) a}$ .

$- \frac{4^{u}}{2 \ln (4) a} + C$

Step 8

Replace all occurrences of $u$ with $- 2 a x$ .

$- \frac{4^{- 2 a x}}{2 \ln (4) a} + C$