Calculus Examples

$\frac{d y}{d x} = c e^{x}$

Step 1

Rewrite the equation.

$d y = c e^{x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int c e^{x} d x$

Step 2.2

Apply the constant rule.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

$y + C_{1} = c (e^{x} + C_{2})$

Step 2.3.3

Simplify.

$y + C_{1} = c e^{x} + C_{2}$

Step 2.3.4

Reorder terms.

$y + C_{1} = e^{x} c + C_{2}$

Step 2.4

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

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