Calculus Examples

$3 \frac{d y}{d x} - 3 e^{x} = 9 x^{2}$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Add $3 e^{x}$ to both sides of the equation.

$3 \frac{d y}{d x} = 9 x^{2} + 3 e^{x}$

Step 1.1.2

Divide each term in $3 \frac{d y}{d x} = 9 x^{2} + 3 e^{x}$ by $3$ and simplify.

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Step 1.1.2.1

Divide each term in $3 \frac{d y}{d x} = 9 x^{2} + 3 e^{x}$ by $3$ .

$\frac{3 \frac{d y}{d x}}{3} = \frac{9 x^{2}}{3} + \frac{3 e^{x}}{3}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $3$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{3 \frac{d y}{d x}}{3} = \frac{9 x^{2}}{3} + \frac{3 e^{x}}{3}$

Step 1.1.2.2.1.2

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

$\frac{d y}{d x} = \frac{9 x^{2}}{3} + \frac{3 e^{x}}{3}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Cancel the common factor of $9$ and $3$ .

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Step 1.1.2.3.1.1.1

Factor $3$ out of $9 x^{2}$ .

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

Step 1.1.2.3.1.1.2

Cancel the common factors.

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Step 1.1.2.3.1.1.2.1

Factor $3$ out of $3$ .

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

Step 1.1.2.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.2.3.1.1.2.4

Divide $3 x^{2}$ by $1$ .

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

Step 1.1.2.3.1.2

Cancel the common factor of $3$ .

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Step 1.1.2.3.1.2.1

Cancel the common factor.

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

Step 1.1.2.3.1.2.2

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

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

Step 1.2

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int 3 x^{2} + e^{x} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 2.3.5.2

Simplify.

$y + C_{1} = x^{3} + e^{x} + C_{4}$

Step 2.4

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

$y = x^{3} + e^{x} + K$