How do you integrate sin²x?
use the identity:
sin(a)sin(b)=(cos(a-b)-cos(a+b))/2
since a=b=x,
sin²(x)
=(cos(x-x)-cos(x+s))/2
=(1-cos(2x))/2
The last expression can be integrated with the standard rules.
To integrate sin²x, you can use a trigonometric identity to simplify the expression. The identity states that sin²x = (1 - cos(2x))/2.
To derive this identity, start with the double angle formula: cos(2x) = cos²x - sin²x. Rearrange the equation to solve for sin²x: sin²x = 1 - cos²x.
Now, substitute sin²x in the integral with the derived identity: ∫ sin²x dx = ∫ (1 - cos(2x))/2 dx.
Next, you can split the integral into two parts: ∫ (1/2 - cos(2x)/2) dx.
Integrate each part separately: ∫ (1/2) dx - ∫ (cos(2x)/2) dx.
The first integral of (1/2) is simply (1/2)x + C, where C is the constant of integration.
For the second integral, use the substitution u = 2x. Then, du = 2dx. Rewriting the integral with this substitution, we have: 1/2 ∫ cos(u) du.
Integrating cos(u) with respect to u gives sin(u): 1/2 (sin(u)) + C.
Substituting back u = 2x, we get 1/2 (sin(2x)) + C.
Therefore, the final answer is (1/2)x - (1/4) sin(2x) + C, where C is the constant of integration.