What is the integral of arccos(cos(x)) ?

The integral of arccos(cos(x)) is 1/2 arccos^2(cos(x))+C

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integral of arccos(cos(x))

\int arccos (cos (x)) d x

\frac{1}{2} arccos^{2} (cos (x)) + C

Solution steps

\int arccos (cos (x)) d x

Apply u-substitution

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

Apply u-substitution

= - \int - v d v

Take the constant out:

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

= - (- \int v d v)

Apply the Power Rule

= - (- \frac{v ^{2}}{2})

Substitute back

= - (- \frac{arccos ^{2} ( cos ( x ) )}{2})

Simplify

- (- \frac{arccos ^{2} ( cos ( x ) )}{2}) : \frac{1}{2} arccos^{2} (cos (x))

= \frac{1}{2} arccos^{2} (cos (x))

Add a constant to the solution

= \frac{1}{2} arccos^{2} (cos (x)) + C

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What is the integral of arccos(cos(x)) ?
The integral of arccos(cos(x)) is 1/2 arccos^2(cos(x))+C