If _0^5^ x1+5^ x(1+ x3x+^2x+^3x3x)dx = k16, then k is equal to ______.

Question

If $\displaystyle\int_0^\pi\frac{5^{\cos x}}{1+5^{\cos x}}\left(1+\cos x\cos3x+\cos^2x+\cos^3x\cos3x\right)dx = \dfrac{k\pi}{16}$, then $k$ is equal to \_\_\_\_\_\_.

Answer: 13

Accepted Answer

By King's rule ($x\to\pi-x$), using $\dfrac{5^{\cos x}}{1+5^{\cos x}}+\dfrac{5^{-\cos x}}{1+5^{-\cos x}}=1$:

$$2I = \int_0^\pi\left(1+\cos x\cos3x+\cos^2x+\cos^3x\cos3x\right)dx$$

By further even/odd symmetry about $x=\pi/2$, this reduces (as detailed in the source, using product-to-sum identities) to

$$2I = \frac{13}4\cdot\frac\pi2+\left[\text{a term vanishing at the limits}\right] \implies I = \frac{13\pi}{16}$$

So $k=13$.

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