Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.1

Add $1$ to both sides of the equation.

$\frac{d y}{d x} = 2 x - \frac{1}{2} + 1$

Step 1.1.2

Write $1$ as a fraction with a common denominator.

$\frac{d y}{d x} = 2 x - \frac{1}{2} + \frac{2}{2}$

Step 1.1.3

Combine the numerators over the common denominator.

$\frac{d y}{d x} = 2 x + \frac{- 1 + 2}{2}$

Step 1.1.4

Add $- 1$ and $2$ .

$\frac{d y}{d x} = 2 x + \frac{1}{2}$

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 2 x + \frac{1}{2} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 2 x + \frac{1}{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 2 x d x + \int \frac{1}{2} d x$

Step 2.3.2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = 2 \int x d x + \int \frac{1}{2} d x$

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

$y + C_{1} = x^{2} + \frac{1}{2} x + C_{4}$

Step 2.4

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

$y = x^{2} + \frac{1}{2} x + K$