Calculus Examples

$\frac{d y}{d x} = {(1 - 2 x)}^{2}$

Step 1

Rewrite the equation.

$d y = {(1 - 2 x)}^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int {(1 - 2 x)}^{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int {(1 - 2 x)}^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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Step 2.3.1.1.1

Differentiate $1 - 2 x$ .

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

Step 2.3.1.1.2

Differentiate.

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Step 2.3.1.1.2.1

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

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

Step 2.3.1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.3.1.1.3

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

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Step 2.3.1.1.3.1

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

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

Step 2.3.1.1.3.2

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

$0 - 2 \cdot 1$

Step 2.3.1.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.3.1.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.3.1.2

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

$y + C_{1} = \int u^{2} \frac{1}{- 2} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

$y + C_{1} = \int u^{2} (- \frac{1}{2}) d u$

Step 2.3.2.2

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

$y + C_{1} = \int - \frac{u^{2}}{2} d u$

Step 2.3.3

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

$y + C_{1} = - \int \frac{u^{2}}{2} d u$

Step 2.3.4

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

$y + C_{1} = - (\frac{1}{2} \int u^{2} d u)$

Step 2.3.5

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

$y + C_{1} = - \frac{1}{2} (\frac{1}{3} u^{3} + C_{2})$

Step 2.3.6

Simplify.

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Step 2.3.6.1

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

$y + C_{1} = - \frac{1}{2} \cdot \frac{1}{3} u^{3} + C_{2}$

Step 2.3.6.2

Simplify.

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Step 2.3.6.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$y + C_{1} = - \frac{1}{3 \cdot 2} u^{3} + C_{2}$

Step 2.3.6.2.2

Multiply $3$ by $2$ .

$y + C_{1} = - \frac{1}{6} u^{3} + C_{2}$

Step 2.3.7

Replace all occurrences of $u$ with $1 - 2 x$ .

$y + C_{1} = - \frac{1}{6} {(1 - 2 x)}^{3} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = - \frac{1}{6} {(1 - 2 x)}^{3} + K$