# Find the real positif number

#### dhiab

Find the real positif number a such that :
$$\displaystyle \int_{0}^{a}\frac{tan(\frac{\pi }{4}+\frac{x}{2})}{sec^{2}(x)}dx=\frac{1}{16}$$

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Hello dhiab
Find the real positif number a such that :
$$\displaystyle \int_{0}^{a}\frac{tan(\frac{\pi }{4}+\frac{x}{2})}{sec^{2}(x)}dx=\frac{1}{16}$$
There are many positive solutions. Here is the first one. But it doesn't work out very neatly, so check my working!

$$\displaystyle \tan\left(\frac{\pi }{4}+\frac{x}{2}\right)=\frac{1+\tan\frac x2}{1-\tan\frac x2}$$
$$\displaystyle =\frac{\cos\frac x2+\sin\frac x2}{\cos\frac x2-\sin\frac x2}$$

$$\displaystyle =\frac{(\cos\frac x2+\sin\frac x2)^2}{\cos^2\frac x2-\sin^2\frac x2}$$

$$\displaystyle =\frac{1+2\sin\frac x2\cos\frac x2}{\cos x}$$

$$\displaystyle =\frac{1+\sin x}{\cos x}$$

$$\displaystyle \Rightarrow \int_0^a\frac{\tan\left(\frac{\pi }{4}+\frac{x}{2}\right)}{\sec^2x}\;dx = \int_0^a(\cos x + \cos x\sin x)\;dx$$
$$\displaystyle =\int_0^a(\cos x +\tfrac12 \sin2x)\;dx$$

$$\displaystyle =\Big[\sin x -\tfrac14\cos2x\Big]_0^a$$

$$\displaystyle =\sin a - \tfrac14\cos2a +\tfrac14$$

$$\displaystyle =\tfrac1{16}$$

$$\displaystyle \Rightarrow 16\sin a -4(1-2\sin^2a)+4=1$$

$$\displaystyle \Rightarrow 8\sin^2a+16\sin a -1=0$$

$$\displaystyle \Rightarrow \sin a = \frac{-16+\sqrt{256+32}}{16}$$, taking the positive root
$$\displaystyle =\frac{-4+\sqrt{18}}{4}$$
So the first positive solution is
$$\displaystyle a = \arcsin\left(\frac{-4+\sqrt{18}}{4}\right)$$