Calculus Examples

$(2 \sin (y) + 1) \frac{d y}{d x} = 4$

Step 1

Rewrite the equation.

$(2 \sin (y) + 1) d y = 4 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 2 \sin (y) + 1 d y = \int 4 d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 2 \sin (y) d y + \int d y = \int 4 d x$

Step 2.2.2

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

$2 \int \sin (y) d y + \int d y = \int 4 d x$

Step 2.2.3

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

$2 (- \cos (y) + C_{1}) + \int d y = \int 4 d x$

Step 2.2.4

Apply the constant rule.

$2 (- \cos (y) + C_{1}) + y + C_{2} = \int 4 d x$

Step 2.2.5

Simplify.

$- 2 \cos (y) + y + C_{3} = \int 4 d x$

Step 2.2.6

Reorder terms.

$y - 2 \cos (y) + C_{3} = \int 4 d x$

Step 2.3

Apply the constant rule.

$y - 2 \cos (y) + C_{3} = 4 x + C_{4}$

Step 2.4

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

$y - 2 \cos (y) = 4 x + K$