The integral of sin^2x is a common mathematical problem that often arises in calculus. To find the integral, we need to apply integration techniques and use trigonometric identities.
First, let’s recall the double-angle identity for sine: sin(2x) = 2sin(x)cos(x). This identity will be useful in simplifying the integral.
To start, we can rewrite sin^2x as (1/2)(1 – cos(2x)). This is a result of using the identity sin^2x = (1/2)(1 – cos(2x)). Now, we can proceed with integrating this expression.
∫ sin^2x dx = ∫ (1/2)(1 – cos(2x)) dx
Using the distributive property of integration, we can split the integral into two separate integrals:
∫ (1/2) dx – ∫ (1/2)cos(2x) dx
The first integral is straightforward:
(1/2) ∫ dx = (1/2)x + C1
For the second integral, we can use a substitution to simplify it. Let u = 2x, then du = 2 dx. Rearranging, we have dx = (1/2) du. Substituting these values into the integral:
∫ (1/2)cos(2x) dx = ∫ cos(u) (1/2) du = (1/2) ∫ cos(u) du
Now, the integral of cos(u) is sin(u):
(1/2) ∫ cos(u) du = (1/2) sin(u) + C2
Since u = 2x, we can substitute back:
(1/2) sin(u) + C2 = (1/2) sin(2x) + C2
Bringing the two integrals together:
∫ sin^2x dx = (1/2)x – (1/2) sin(2x) + C
We can simplify the expression:
∫ sin^2x dx = -(1/2) sin(2x) + (1/2)x + C
As a result, the integral of sin^2x is given by -(1/2) sin(2x) + (1/2)x + C, where C represents the constant of integration.
It is worth noting that integration can have various applications in real-life situations. For example, in physics, integration is used to calculate areas, volumes, and displacements. In finance, integration is employed to determine the total profit or loss over a given period. These applications demonstrate the practical significance of understanding integration techniques and formulas like the one we discussed here.