Calculus Examples

$\frac{d y}{d x} = \sin (x) - \cos (x)$

Step 1

Rewrite the equation.

$d y = (\sin (x) - \cos (x)) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \sin (x) - \cos (x) d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \sin (x) - \cos (x) d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

$y + C_{1} = \int \sin (x) d x + \int - \cos (x) d x$

Step 2.3.2

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$y + C_{1} = - \cos (x) + C_{2} + \int - \cos (x) d x$

Step 2.3.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$y + C_{1} = - \cos (x) + C_{2} - \int \cos (x) d x$

Step 2.3.4

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$y + C_{1} = - \cos (x) + C_{2} - (\sin (x) + C_{3})$

Step 2.3.5

Simplify.

$y + C_{1} = - \cos (x) - \sin (x) + C_{4}$

Step 2.4

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

$y = - \cos (x) - \sin (x) + K$