Calculus Examples

$4 x + \frac{d y}{d x} = 36 x^{5} + 12$

Step 1

Separate the variables.

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$\frac{d y}{d x} = 36 x^{5} + 12 - 4 x$

Step 1.2

Rewrite the equation.

$d y = (36 x^{5} + 12 - 4 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 36 x^{5} + 12 - 4 x d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int 36 x^{5} + 12 - 4 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int 36 x^{5} d x + \int 12 d x + \int - 4 x d x$

Step 2.3.2

Since $36$ is constant with respect to $x$ , move $36$ out of the integral.

$y + C_{1} = 36 \int x^{5} d x + \int 12 d x + \int - 4 x d x$

Step 2.3.3

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$y + C_{1} = 36 (\frac{1}{6} x^{6} + C_{2}) + \int 12 d x + \int - 4 x d x$

Step 2.3.4

Apply the constant rule.

$y + C_{1} = 36 (\frac{1}{6} x^{6} + C_{2}) + 12 x + C_{3} + \int - 4 x d x$

Step 2.3.5

Combine $\frac{1}{6}$ and $x^{6}$ .

$y + C_{1} = 36 (\frac{x^{6}}{6} + C_{2}) + 12 x + C_{3} + \int - 4 x d x$

Step 2.3.6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$y + C_{1} = 36 (\frac{x^{6}}{6} + C_{2}) + 12 x + C_{3} - 4 \int x d x$

Step 2.3.7

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

$y + C_{1} = 36 (\frac{x^{6}}{6} + C_{2}) + 12 x + C_{3} - 4 (\frac{1}{2} x^{2} + C_{4})$

Step 2.3.8

Simplify.

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Step 2.3.8.1

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

$y + C_{1} = 36 (\frac{x^{6}}{6} + C_{2}) + 12 x + C_{3} - 4 (\frac{x^{2}}{2} + C_{4})$

Step 2.3.8.2

Simplify.

$y + C_{1} = 6 x^{6} + 12 x - 4 \frac{x^{2}}{2} + C_{5}$

Step 2.3.8.3

Simplify.

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Step 2.3.8.3.1

Combine $- 4$ and $\frac{x^{2}}{2}$ .

$y + C_{1} = 6 x^{6} + 12 x + \frac{- 4 x^{2}}{2} + C_{5}$

Step 2.3.8.3.2

Cancel the common factor of $- 4$ and $2$ .

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Step 2.3.8.3.2.1

Factor $2$ out of $- 4 x^{2}$ .

$y + C_{1} = 6 x^{6} + 12 x + \frac{2 (- 2 x^{2})}{2} + C_{5}$

Step 2.3.8.3.2.2

Cancel the common factors.

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Step 2.3.8.3.2.2.1

Factor $2$ out of $2$ .

$y + C_{1} = 6 x^{6} + 12 x + \frac{2 (- 2 x^{2})}{2 (1)} + C_{5}$

Step 2.3.8.3.2.2.2

Cancel the common factor.

$y + C_{1} = 6 x^{6} + 12 x + \frac{2 (- 2 x^{2})}{2 \cdot 1} + C_{5}$

Step 2.3.8.3.2.2.3

Rewrite the expression.

$y + C_{1} = 6 x^{6} + 12 x + \frac{- 2 x^{2}}{1} + C_{5}$

Step 2.3.8.3.2.2.4

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

$y + C_{1} = 6 x^{6} + 12 x - 2 x^{2} + C_{5}$

Step 2.3.9

Reorder terms.

$y + C_{1} = 6 x^{6} - 2 x^{2} + 12 x + C_{5}$

Step 2.4

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

$y = 6 x^{6} - 2 x^{2} + 12 x + K$