Integration problem involving $\sqrt{\cos(x)-\cos^3(x)}$

Question

Evaluate $$\int_{-\pi/2}^{\pi/2}\sqrt{\cos(x)-\cos^3(x)}\,dx.$$ Please help

Answer 1

Use $$\cos{x}-\cos^3x=\sin^2x\cos{x}.$$

We obtain: $$\int\limits_{0}^{\frac{\pi}{2}}\sin{x}\sqrt{\cos{x}}dx+\int_{-\frac{\pi}{2}}^0\left(-\sin{x}\sqrt{\cos{x}}\right)dx=-2\int\limits_{0}^{\frac{\pi}{2}}\sqrt{\cos{x}}d(\cos{x})=$$ $$=-2\frac{\sqrt{\cos^3x}}{\frac{3}{2}}|_0^{\frac{\pi}{2}}=\frac{4}{3}.$$

Answer 2

You are correct. Notice that for $x\in [-\pi/2,\pi/2]$, $$\sqrt{\cos(x)-\cos^3(x)}=\sqrt{\cos(x)(1-\cos^2(x))}=|\sin(x)|\sqrt{\cos(x)}$$ Hence $$\int_{-\pi/2}^{\pi/2}\sqrt{\cos(x)-\cos^3(x)}dx= 2\int_{0}^{\pi/2}\sqrt{\cos(x)}\sin(x)dx=-\frac{4}{3}\left[\cos(x)^{3/2}\right]_0^{\pi/2}=\frac{4}{3}.$$

Answer 3

for the indefinite integral we get $$-\frac{2}{3}\frac{\cos(x)\sqrt{\cos(x)\sin^2(x)}}{\sin(x)}+C$$