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

The integral of sqrt(cos(x)-cos^3(x)) is -2/3 cos^{3/2}(x)+C

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

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

- \frac{2}{3} cos^{\frac{3}{2}} (x) + C

Solution steps

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

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = cos (x)

= - \frac{2}{3} cos^{\frac{3}{2}} (x)

Add a constant to the solution

= - \frac{2}{3} cos^{\frac{3}{2}} (x) + C

f (x) = 7 + x^{2}

7 e^{7 e^{x} + x}

f (z) = 3 x + 2 z + 4 x z

f (x) = 12 x^{4} + 3 x^{3} + 5 x^{- 2} - 4

\int sin^{2} (b x) cos (b x) d x

What is the integral of sqrt(cos(x)-cos^3(x)) ?
The integral of sqrt(cos(x)-cos^3(x)) is -2/3 cos^{3/2}(x)+C