The value of _0^/2(^3x)/( x+ x),dx is:

Question

The value of $\displaystyle\int_0^{\pi/2}\frac{\sin^3x}{\sin x+\cos x}\,dx$ is:

Accepted Answer

Let $I=\displaystyle\int_0^{\pi/2}\frac{\sin^3x}{\sin x+\cos x}dx\ \ldots(1)$. Using $x\to\dfrac\pi2-x$:
$$I=\int_0^{\pi/2}\frac{\cos^3x}{\sin x+\cos x}dx\ \ldots(2)$$

Adding (1) and (2):
$$2I=\int_0^{\pi/2}\frac{\sin^3x+\cos^3x}{\sin x+\cos x}dx=\int_0^{\pi/2}(1-\sin x\cos x)\,dx=\int_0^{\pi/2}\left(1-\frac12\sin2x\right)dx$$

$$2I=\left[x+\frac14\cos2x\right]_0^{\pi/2}=\frac\pi2-1\Rightarrow I=\frac{\pi-1}4$$

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