Calculus Examples

$\frac{6 x}{2^{4 x^{2}}}$

Step 1

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{x}{2^{4 x^{2}}} d x$

Step 2

Simplify the expression.

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Step 2.1

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

$6 \int x {(2^{4 x^{2}})}^{- 1} d x$

Step 2.2

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

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Step 2.2.1

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

$6 \int x \cdot 2^{4 x^{2} \cdot - 1} d x$

Step 2.2.2

Multiply $- 1$ by $4$ .

$6 \int x \cdot 2^{- 4 x^{2}} d x$

Step 3

Let $u = - 4 x^{2}$ . Then $d u = - 8 x d x$ , so $- \frac{1}{8} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

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

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Step 3.1.1

Differentiate $- 4 x^{2}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- 4 (2 x)$

Step 3.1.4

Multiply $2$ by $- 4$ .

$- 8 x$

Step 3.2

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

$6 \int 2^{u} \frac{1}{- 8} d u$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$6 \int 2^{u} (- \frac{1}{8}) d u$

Step 4.2

Combine $2^{u}$ and $\frac{1}{8}$ .

$6 \int - \frac{2^{u}}{8} d u$

Step 5

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

$6 (- \int \frac{2^{u}}{8} d u)$

Step 6

Multiply $- 1$ by $6$ .

$- 6 \int \frac{2^{u}}{8} d u$

Step 7

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$- 6 (\frac{1}{8} \int 2^{u} d u)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{8}$ and $- 6$ .

$\frac{- 6}{8} \int 2^{u} d u$

Step 8.2

Cancel the common factor of $- 6$ and $8$ .

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Step 8.2.1

Factor $2$ out of $- 6$ .

$\frac{2 (- 3)}{8} \int 2^{u} d u$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot - 3}{2 \cdot 4} \int 2^{u} d u$

Step 8.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 3}{2 \cdot 4} \int 2^{u} d u$

Step 8.2.2.3

Rewrite the expression.

$\frac{- 3}{4} \int 2^{u} d u$

Step 8.3

Move the negative in front of the fraction.

$- \frac{3}{4} \int 2^{u} d u$

Step 9

The integral of $2^{u}$ with respect to $u$ is $\frac{2^{u}}{\ln (2)}$ .

$- \frac{3}{4} (\frac{2^{u}}{\ln (2)} + C)$

Step 10

Simplify.

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Step 10.1

Rewrite $- \frac{3}{4} (\frac{2^{u}}{\ln (2)} + C)$ as $- \frac{3}{4} \cdot \frac{1}{\ln (2)} \cdot 2^{u} + C$ .

$- \frac{3}{4} \cdot \frac{1}{\ln (2)} \cdot 2^{u} + C$

Step 10.2

Simplify.

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Step 10.2.1

Multiply $\frac{1}{\ln (2)}$ by $\frac{3}{4}$ .

$- \frac{3}{\ln (2) \cdot 4} \cdot 2^{u} + C$

Step 10.2.2

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

$- \frac{3}{4 \ln (2)} \cdot 2^{u} + C$

Step 11

Replace all occurrences of $u$ with $- 4 x^{2}$ .

$- \frac{3}{4 \ln (2)} \cdot 2^{- 4 x^{2}} + C$