What is the integral of (3-cos(x))^2 ?

The integral of (3-cos(x))^2 is -6sin(x)-19/2 arcsin(cos(x))+1/4 sin(2arcsin(cos(x)))+C

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integral of (3-cos(x))^2

\int (3 - cos (x))^{2} d x

- 6 sin (x) - \frac{19}{2} arcsin (cos (x)) + \frac{1}{4} sin (2 arcsin (cos (x))) + C

Solution steps

\int (3 - cos (x))^{2} d x

Factor

3 - cos (x) : - (- 3 + cos (x))

= \int (- (- 3 + cos (x)))^{2} d x

Simplify

(- (- 3 + cos (x)))^{2} : (- 3 + cos (x))^{2}

= \int (- 3 + cos (x))^{2} d x

Apply u-substitution

= \int - \frac{( - 3 + u ) ^{2}}{1 - u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{( - 3 + u ) ^{2}}{1 - u ^{2}} d u

Expand

\frac{( - 3 + u ) ^{2}}{1 - u ^{2}} : \frac{9}{1 - u ^{2}} - \frac{6 u}{1 - u ^{2}} + \frac{u ^{2}}{1 - u ^{2}}

= - \int \frac{9}{1 - u ^{2}} - \frac{6 u}{1 - u ^{2}} + \frac{u ^{2}}{1 - u ^{2}} d u

Apply the Sum Rule:

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

= - (\int \frac{9}{1 - u ^{2}} d u - \int \frac{6 u}{1 - u ^{2}} d u + \int \frac{u ^{2}}{1 - u ^{2}} d u)

\int \frac{9}{1 - u ^{2}} d u = 9 arcsin (u)

\int \frac{6 u}{1 - u ^{2}} d u = - 6 1 - u^{2}

\int \frac{u ^{2}}{1 - u ^{2}} d u = \frac{1}{2} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u)))

= - (9 arcsin (u) - (- 6 1 - u^{2}) + \frac{1}{2} (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))))

Substitute back

u = cos (x)

= - (9 arcsin (cos (x)) - (- 6 1 - cos^{2} (x)) + \frac{1}{2} (arcsin (cos (x)) - \frac{1}{2} sin (2 arcsin (cos (x)))))

Simplify

- (9 arcsin (cos (x)) - (- 6 1 - cos^{2} (x)) + \frac{1}{2} (arcsin (cos (x)) - \frac{1}{2} sin (2 arcsin (cos (x))))) : - 6 sin (x) - \frac{19}{2} arcsin (cos (x)) + \frac{1}{4} sin (2 arcsin (cos (x)))

= - 6 sin (x) - \frac{19}{2} arcsin (cos (x)) + \frac{1}{4} sin (2 arcsin (cos (x)))

Add a constant to the solution

= - 6 sin (x) - \frac{19}{2} arcsin (cos (x)) + \frac{1}{4} sin (2 arcsin (cos (x))) + C

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What is the integral of (3-cos(x))^2 ?
The integral of (3-cos(x))^2 is -6sin(x)-19/2 arcsin(cos(x))+1/4 sin(2arcsin(cos(x)))+C