Calculus Examples

$\frac{d y}{d x} + (\frac{1}{x}) y = 6 x + 2$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = \frac{1}{x}$ .

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Step 1.1

Set up the integration.

$e^{\int \frac{1}{x} d x}$

Step 1.2

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

$e^{\ln (| x |) + C}$

Step 1.3

Remove the constant of integration.

$e^{\ln (x)}$

Step 1.4

Exponentiation and log are inverse functions.

$x$

Step 2

Multiply each term by the integrating factor $x$ .

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Step 2.1

Multiply each term by $x$ .

$x \frac{d y}{d x} + x ((\frac{1}{x}) y) = x (6 x) + x \cdot 2$

Step 2.2

Simplify each term.

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Step 2.2.1

Combine $\frac{1}{x}$ and $y$ .

$x \frac{d y}{d x} + x \frac{y}{x} = x (6 x) + x \cdot 2$

Step 2.2.2

Cancel the common factor of $x$ .

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Step 2.2.2.1

Cancel the common factor.

$x \frac{d y}{d x} + x \frac{y}{x} = x (6 x) + x \cdot 2$

Step 2.2.2.2

Rewrite the expression.

$x \frac{d y}{d x} + y = x (6 x) + x \cdot 2$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite using the commutative property of multiplication.

$x \frac{d y}{d x} + y = 6 x \cdot x + x \cdot 2$

Step 2.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.3.2.1

Move $x$ .

$x \frac{d y}{d x} + y = 6 (x \cdot x) + x \cdot 2$

Step 2.3.2.2

Multiply $x$ by $x$ .

$x \frac{d y}{d x} + y = 6 x^{2} + x \cdot 2$

Step 2.3.3

Move $2$ to the left of $x$ .

$x \frac{d y}{d x} + y = 6 x^{2} + 2 x$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [x y] = 6 x^{2} + 2 x$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [x y] d x = \int 6 x^{2} + 2 x d x$

Step 5

Integrate the left side.

$x y = \int 6 x^{2} + 2 x d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$x y = \int 6 x^{2} d x + \int 2 x d x$

Step 6.2

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

$x y = 6 \int x^{2} d x + \int 2 x d x$

Step 6.3

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

$x y = 6 (\frac{1}{3} x^{3} + C) + \int 2 x d x$

Step 6.4

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

$x y = 6 (\frac{1}{3} x^{3} + C) + 2 \int x d x$

Step 6.5

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

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

Step 6.6

Simplify.

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Step 6.6.1

Simplify.

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

Step 6.6.2

Simplify.

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Step 6.6.2.1

Combine $6$ and $\frac{1}{3}$ .

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

Step 6.6.2.2

Cancel the common factor of $6$ and $3$ .

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Step 6.6.2.2.1

Factor $3$ out of $6$ .

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

Step 6.6.2.2.2

Cancel the common factors.

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Step 6.6.2.2.2.1

Factor $3$ out of $3$ .

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

Step 6.6.2.2.2.2

Cancel the common factor.

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

Step 6.6.2.2.2.3

Rewrite the expression.

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

Step 6.6.2.2.2.4

Divide $2$ by $1$ .

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

Step 6.6.2.3

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

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

Step 6.6.2.4

Cancel the common factor of $2$ .

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Step 6.6.2.4.1

Cancel the common factor.

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

Step 6.6.2.4.2

Rewrite the expression.

$x y = 2 x^{3} + 1 x^{2} + C$

Step 6.6.2.5

Multiply $x^{2}$ by $1$ .

$x y = 2 x^{3} + x^{2} + C$

Step 7

Divide each term in $x y = 2 x^{3} + x^{2} + C$ by $x$ and simplify.

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Step 7.1

Divide each term in $x y = 2 x^{3} + x^{2} + C$ by $x$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $x$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $y$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Cancel the common factor of $x^{3}$ and $x$ .

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Step 7.3.1.1.1

Factor $x$ out of $2 x^{3}$ .

$y = \frac{x (2 x^{2})}{x} + \frac{x^{2}}{x} + \frac{C}{x}$

Step 7.3.1.1.2

Cancel the common factors.

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Step 7.3.1.1.2.1

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

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

Step 7.3.1.1.2.2

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

$y = \frac{x (2 x^{2})}{x \cdot 1} + \frac{x^{2}}{x} + \frac{C}{x}$

Step 7.3.1.1.2.3

Cancel the common factor.

$y = \frac{x (2 x^{2})}{x \cdot 1} + \frac{x^{2}}{x} + \frac{C}{x}$

Step 7.3.1.1.2.4

Rewrite the expression.

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

Step 7.3.1.1.2.5

Divide $2 x^{2}$ by $1$ .

$y = 2 x^{2} + \frac{x^{2}}{x} + \frac{C}{x}$

Step 7.3.1.2

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

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Step 7.3.1.2.1

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

$y = 2 x^{2} + \frac{x \cdot x}{x} + \frac{C}{x}$

Step 7.3.1.2.2

Cancel the common factors.

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Step 7.3.1.2.2.1

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

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

Step 7.3.1.2.2.2

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

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

Step 7.3.1.2.2.3

Cancel the common factor.

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

Step 7.3.1.2.2.4

Rewrite the expression.

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

Step 7.3.1.2.2.5

Divide $x$ by $1$ .

$y = 2 x^{2} + x + \frac{C}{x}$