K = \int (1 - \cos x)^2 \, dx

To solve the integral K = ∫ ( 1 − cos x ) 2 d x , we can use trigonometric identities and substitution methods to simplify and integrate.
Here's a step-by-step guide to solve it:

Expand the Expression : To handle the integral ∫ ( 1 − cos x ) 2 d x , first expand the squared term:
( 1 − cos x ) 2 = 1 − 2 cos x + cos 2 x .

Break Down the Integral : Using the expanded expression, the integral becomes:
K = ∫ ( 1 − 2 cos x + cos 2 x ) d x = ∫ 1 d x − ∫ 2 cos x d x + ∫ cos 2 x d x .

Integrate Each Term :

The integral of 1 is straightforward: ∫ 1 d x = x .
The integral of 2 cos x is similarly straightforward: ∫ 2 cos x d x = 2 sin x .
For ∫ cos 2 x d x , use the power-reduction identity: cos 2 x = 2 1 + cos 2 x ​ . Substituting this identity, we have: ∫ cos 2 x d x = ∫ 2 1 + cos 2 x ​ d x = 2 1 ​ ∫ 1 d x + 2 1 ​ ∫ cos 2 x d x .
This further simplifies to: 2 1 ​ x + 4 1 ​ sin 2 x .



Putting it all together, we get:

Combine the Results :

Combine everything: K = x − 2 sin x + 2 1 ​ x + 4 1 ​ sin 2 x + C , where C is the constant of integration.


Simplify the Expression :

Combine like terms: K = 2 3 ​ x − 2 sin x + 4 1 ​ sin 2 x + C .



Thus, the integral ∫ ( 1 − cos x ) 2 d x evaluates to: K = 2 3 ​ x − 2 sin x + 4 1 ​ sin 2 x + C .

Answered by LiamAlexanderSmith | 2025-07-06