Calculus Examples

$(\cos (y) + 2) \frac{d y}{d x} = 2 x$

Step 1

Rewrite the equation.

$(\cos (y) + 2) d y = 2 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \cos (y) + 2 d y = \int 2 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int \cos (y) d y + \int 2 d y = \int 2 x d x$

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

$\sin (y) + 2 y + C_{3} = \int 2 x d x$

Step 2.2.5

Reorder terms.

$2 y + \sin (y) + C_{3} = \int 2 x d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 y + \sin (y) + C_{3} = 2 \int x d x$

Step 2.3.2

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

$2 y + \sin (y) + C_{3} = 2 (\frac{1}{2} x^{2} + C_{4})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $2 (\frac{1}{2} x^{2} + C_{4})$ as $2 (\frac{1}{2}) x^{2} + C_{4}$ .

$2 y + \sin (y) + C_{3} = 2 (\frac{1}{2}) x^{2} + C_{4}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

$2 y + \sin (y) + C_{3} = \frac{2}{2} x^{2} + C_{4}$

Step 2.3.3.2.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.2.1

Cancel the common factor.

$2 y + \sin (y) + C_{3} = \frac{2}{2} x^{2} + C_{4}$

Step 2.3.3.2.2.2

Rewrite the expression.

$2 y + \sin (y) + C_{3} = 1 x^{2} + C_{4}$

Step 2.3.3.2.3

Multiply $x^{2}$ by $1$ .

$2 y + \sin (y) + C_{3} = x^{2} + C_{4}$

Step 2.4

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

$2 y + \sin (y) = x^{2} + K$