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

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

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

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

3 x + 4 cos (x) - sin (2 x) + C

Solution steps

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

Factor

1 - 2 sin (x) : - (- 1 + 2 sin (x))

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

Simplify

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

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

Expand

(- 1 + 2 sin (x))^{2} : 1 - 4 sin (x) + 4 sin^{2} (x)

= \int 1 - 4 sin (x) + 4 sin^{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 4 sin (x) d x + \int 4 sin^{2} (x) d x

\int 1 d x = x

\int 4 sin (x) d x = - 4 cos (x)

\int 4 sin^{2} (x) d x = 2 x - sin (2 x)

= x - (- 4 cos (x)) + 2 x - sin (2 x)

Simplify

= 3 x + 4 cos (x) - sin (2 x)

Add a constant to the solution

= 3 x + 4 cos (x) - sin (2 x) + C

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