What is the integral of (4-4cos(x))^2 ?

The integral of (4-4cos(x))^2 is 24x-32sin(x)+4sin(2x)+C

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integral of (4-4cos(x))^2

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

24 x - 32 sin (x) + 4 sin (2 x) + C

Solution steps

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

Expand

(4 - 4 cos (x))^{2} : 16 - 32 cos (x) + 16 cos^{2} (x)

= \int 16 - 32 cos (x) + 16 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 16 d x - \int 32 cos (x) d x + \int 16 cos^{2} (x) d x

\int 16 d x = 16 x

\int 32 cos (x) d x = 32 sin (x)

\int 16 cos^{2} (x) d x = 8 (x + \frac{1}{2} sin (2 x))

= 16 x - 32 sin (x) + 8 (x + \frac{1}{2} sin (2 x))

Simplify

16 x - 32 sin (x) + 8 (x + \frac{1}{2} sin (2 x)) : 24 x - 32 sin (x) + 4 sin (2 x)

= 24 x - 32 sin (x) + 4 sin (2 x)

Add a constant to the solution

= 24 x - 32 sin (x) + 4 sin (2 x) + C

y = 4 x - x^{2}, at x = a

\frac{d}{d x} (2 x (4 x + 9)^{\frac{1}{2}} + 12)

\frac{d}{d x} ((x^{2} + x) \cdot (tan (x)))

\frac{d}{d x} (arcsin (- x))

\frac{d}{d x} (6 sec (x^{2}))

What is the integral of (4-4cos(x))^2 ?
The integral of (4-4cos(x))^2 is 24x-32sin(x)+4sin(2x)+C