What is the integral of (1-cos(x))^2 ?

The integral of (1-cos(x))^2 is 3/2 x+1/4 sin(2x)-2sin(x)+C

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integral of (1-cos(x))^2

\int (1 - cos (x))^{2} d x

\frac{3}{2} x + \frac{1}{4} sin (2 x) - 2 sin (x) + C

Solution steps

\int (1 - cos (x))^{2} d x

Expand

(1 - cos (x))^{2} : 1 - 2 cos (x) + cos^{2} (x)

= \int 1 - 2 cos (x) + cos^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x - \int 2 cos (x) d x + \int cos^{2} (x) d x

\int 1 d x = x

\int 2 cos (x) d x = 2 sin (x)

\int cos^{2} (x) d x = \frac{1}{2} (x + \frac{1}{2} sin (2 x))

= x - 2 sin (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x))

Simplify

x - 2 sin (x) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : \frac{3}{2} x + \frac{1}{4} sin (2 x) - 2 sin (x)

= \frac{3}{2} x + \frac{1}{4} sin (2 x) - 2 sin (x)

Add a constant to the solution

= \frac{3}{2} x + \frac{1}{4} sin (2 x) - 2 sin (x) + C

4 - 4 \cdot e^{2 t}

f (x) = \frac{e ^{x}}{x ^{2} + 1}

\int (7 x^{2}) e^{x^{3} + 1} d x

\int e^{- 2 y} y d y

f (x) = - x^{3} + x^{2} + 4 x - 2, at x = 0

What is the integral of (1-cos(x))^2 ?
The integral of (1-cos(x))^2 is 3/2 x+1/4 sin(2x)-2sin(x)+C