Calculus Examples

$\frac{d}{d x} (\frac{d y}{d x}) = 6 x + 3$

Step 1

Rewrite the equation.

$\frac{d}{d x} d y = (6 x + 3) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{d}{d x} d y = \int 6 x + 3 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply the constant rule.

$\frac{d}{d x} y + C_{1} = \int 6 x + 3 d x$

Step 2.2.2

Simplify the answer.

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Step 2.2.2.1

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

$\frac{d y}{d x} + C_{1} = \int 6 x + 3 d x$

Step 2.2.2.2

Rewrite $\frac{d y}{d x} + C_{1}$ as $\frac{1}{d x} d y + C_{1}$ .

$\frac{1}{d x} d y + C_{1} = \int 6 x + 3 d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$\frac{1}{d x} d y + C_{1} = \int 6 x d x + \int 3 d x$

Step 2.3.2

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

$\frac{1}{d x} d y + C_{1} = 6 \int x d x + \int 3 d x$

Step 2.3.3

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

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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

Step 2.3.5.2

Simplify.

$\frac{1}{d x} d y + C_{1} = 3 x^{2} + 3 x + C_{4}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Simplify $\frac{1}{d x} d y$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

Multiply both sides by $d x$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Cancel the common factor of $d x$ .

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Step 3.3.1.1.1

Cancel the common factor.

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

Step 3.3.1.1.2

Rewrite the expression.

$d y = (3 x^{2} + 3 x + K) d x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $(3 x^{2} + 3 x + K) d x$ .

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Step 3.3.2.1.1

Apply the distributive property.

$d y = 3 x^{2} d x + 3 x d x + K d x$

Step 3.3.2.1.2

Reorder.

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Step 3.3.2.1.2.1

Move $x^{2}$ .

$d y = 3 d x x^{2} + 3 x d x + K d x$

Step 3.3.2.1.2.2

Move $x$ .

$d y = 3 d x x^{2} + 3 d x x + K d x$

Step 3.3.2.1.2.3

Move $3 d x x$ .

$d y = 3 d x x^{2} + K d x + 3 d x x$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $d$ .

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Step 3.4.2.1.1

Cancel the common factor.

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

Step 3.4.2.1.2

Divide $y$ by $1$ .

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

Step 4

Simplify the constant of integration.

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