Calculus Examples

$x \frac{d y}{d x} = x^{2} + 2 x - 3$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x \frac{d y}{d x} = x^{2} + 2 x - 3$ by $x$ and simplify.

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Step 1.1.1

Divide each term in $x \frac{d y}{d x} = x^{2} + 2 x - 3$ by $x$ .

$\frac{x \frac{d y}{d x}}{x} = \frac{x^{2}}{x} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x \frac{d y}{d x}}{x} = \frac{x^{2}}{x} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{x^{2}}{x} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 1.1.3.1.1.1

Factor $x$ out of $x^{2}$ .

$\frac{d y}{d x} = \frac{x \cdot x}{x} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

Raise $x$ to the power of $1$ .

$\frac{d y}{d x} = \frac{x \cdot x}{x^{1}} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.1.2.2

Factor $x$ out of $x^{1}$ .

$\frac{d y}{d x} = \frac{x \cdot x}{x \cdot 1} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.1.2.3

Cancel the common factor.

$\frac{d y}{d x} = \frac{x \cdot x}{x \cdot 1} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.1.2.4

Rewrite the expression.

$\frac{d y}{d x} = \frac{x}{1} + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.1.2.5

Divide $x$ by $1$ .

$\frac{d y}{d x} = x + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.2

Cancel the common factor of $x$ .

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Step 1.1.3.1.2.1

Cancel the common factor.

$\frac{d y}{d x} = x + \frac{2 x}{x} + \frac{- 3}{x}$

Step 1.1.3.1.2.2

Divide $2$ by $1$ .

$\frac{d y}{d x} = x + 2 + \frac{- 3}{x}$

Step 1.1.3.1.3

Move the negative in front of the fraction.

$\frac{d y}{d x} = x + 2 - \frac{3}{x}$

Step 1.2

Rewrite the equation.

$d y = (x + 2 - \frac{3}{x}) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int x + 2 - \frac{3}{x} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int x + 2 - \frac{3}{x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int x d x + \int 2 d x + \int - \frac{3}{x} d x$

Step 2.3.2

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

$y + C_{1} = \frac{1}{2} x^{2} + C_{2} + \int 2 d x + \int - \frac{3}{x} d x$

Step 2.3.3

Apply the constant rule.

$y + C_{1} = \frac{1}{2} x^{2} + C_{2} + 2 x + C_{3} + \int - \frac{3}{x} d x$

Step 2.3.4

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

$y + C_{1} = \frac{1}{2} x^{2} + C_{2} + 2 x + C_{3} - \int \frac{3}{x} d x$

Step 2.3.5

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.6

Multiply $3$ by $- 1$ .

$y + C_{1} = \frac{1}{2} x^{2} + C_{2} + 2 x + C_{3} - 3 \int \frac{1}{x} d x$

Step 2.3.7

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$y + C_{1} = \frac{1}{2} x^{2} + C_{2} + 2 x + C_{3} - 3 (\ln (| x |) + C_{4})$

Step 2.3.8

Simplify.

$y + C_{1} = \frac{1}{2} x^{2} + 2 x - 3 \ln (| x |) + C_{5}$

Step 2.4

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

$y = \frac{1}{2} x^{2} + 2 x - 3 \ln (| x |) + C$