integration of4sin^2ydy

To integrate the expression: ∫4sin^2(y)dy, you will need to use the trigonometric identities and integration techniques. Here's a step-by-step breakdown of the process:

Step 1: Simplify the expression using the trigonometric identity: sin^2(y) = (1/2)(1 - cos(2y))

∫4sin^2(y)dy = ∫4(1/2)(1 - cos(2y))dy

Simplifying further:

∫2(1 - cos(2y))dy

Step 2: Distribute the coefficient of 2 to each term inside the integral:

∫ 2dy - 2(cos(2y))dy

Integrating each term separately:

∫ 2dy = 2y + C1 (where C1 is the constant of integration)

∫ -2(cos(2y))dy = -2∫cos(2y)dy

Step 3: Apply the integration rule for cosine:

Recall that the integral of cos(u) with respect to u is sin(u), so we will substitute u = 2y:

-2∫cos(2y)dy = -2 * (1/2) * sin(2y) + C2

Simplifying:

-2 * (1/2) * sin(2y) + C2 = -sin(2y) + C2

Where C2 is the constant of integration.

Step 4: Combine the results from steps 2 and 3:

Finally, integrating the simplified terms:

∫4sin^2(y)dy = 2y - sin(2y) + C

Where C = C1 + C2 is the combined constant of integration.

So, the integral of 4sin^2(y)dy is 2y - sin(2y) + C, where C is the constant of integration.