Question

$\color{red} \displaystyle \rm \int \frac{1}{x} \prod^{ \infty }_{i = 1} \left( 1 - \tan^{2} \bigg ( \frac{x}{ {2}^{i} } \bigg)\right) \: dx$​

Answer 1

Step-by-step explanation:

We have,

$\displaystyle \rm \int \frac{1}{x} \prod^{ \infty }_{i = 1} \left \{1 - \tan^{2} \left( \frac{x}{ {2}^{i} } \right)\right \}\: dx$

$\displaystyle \rm{ = \int \frac{1}{x} \prod^{ \infty }_{i = 1} \left \{ \dfrac{\cos^{2} \left( \dfrac{x}{ {2}^{i} } \right) - \sin^{2} \left( \dfrac{x}{ {2}^{i} } \right)}{\cos^{2} \left( \dfrac{x}{ {2}^{i} } \right)}\right \}\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \prod^{ \infty }_{i = 1} \left \{ \dfrac{\cos \left( 2 \cdot\dfrac{x}{ {2}^{i} } \right) }{\cos^{2} \left( \dfrac{x}{ {2}^{i} } \right)}\right \}\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \prod^{ \infty }_{i = 1} \left \{ \dfrac{\cos \left( \dfrac{x}{ {2}^{i - 1} } \right) }{\cos^{2} \left( \dfrac{x}{ {2}^{i} } \right)}\right \}\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot \lim_{n \to \infty} \dfrac{\cos \left( \dfrac{x}{ {2}^{1- 1} } \right) \cdot\cos \left( \dfrac{x}{ {2}^{2- 1} } \right) \cdot\cos \left( \dfrac{x}{ {2}^{3- 1} } \right) \cdots\cos \left( \dfrac{x}{ {2}^{n- 1} } \right)}{\cos^{2} \left( \dfrac{x}{ {2}^{1} } \right) \cdot\cos^{2} \left( \dfrac{x}{ {2}^{2} } \right) \cdot\cos^{2} \left( \dfrac{x}{ {2}^{3} } \right) \cdots\cos^{2} \left( \dfrac{x}{ {2}^{n} } \right)}\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot \lim_{n \to \infty}\dfrac{\cos \left( \dfrac{x}{ {2}^{0} } \right) \cdot\cos \left( \dfrac{x}{ {2}^{1} } \right) \cdot\cos \left( \dfrac{x}{ {2}^{2} } \right) \cdots\cos \left( \dfrac{x}{ {2}^{n- 1} } \right)}{ \left \{\cos \left( \dfrac{x}{ {2}^{1} } \right) \cdot\cos\left( \dfrac{x}{ {2}^{2} } \right) \cdot\cos \left( \dfrac{x}{ {2}^{3} } \right) \cdots\cos\left( \dfrac{x}{ {2}^{n} } \right) \right \}^{2} }\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot \lim_{n \to \infty}\dfrac{\cos \left( \dfrac{x}{ {2}^{n - 1} } \right) \cdot\cos \left( 2 \cdot\dfrac{x}{ {2}^{n - 1} } \right) \cdot\cos \left( {2}^{2} \cdot\dfrac{x}{ {2}^{n - 1} } \right) \cdots\cos \left( {2}^{n - 1} \cdot\dfrac{x}{ {2}^{n- 1} } \right)}{ \left \{\cos \left( \dfrac{x}{ {2}^{n} } \right) \cdot\cos\left(2 \cdot \dfrac{x}{ {2}^{n} } \right) \cdot\cos \left( {2}^{2} \cdot \dfrac{x}{ {2}^{n} } \right) \cdots\cos\left( {2}^{n - 1} \cdot\dfrac{x}{ {2}^{n} } \right) \right \}^{2} }\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot\lim_{n \to \infty} \dfrac{ \dfrac{\sin \left( {2}^{n} \cdot \dfrac{x}{ {2}^{n - 1} } \right)}{ {2}^{n} \cdot\sin \left( \dfrac{x}{ {2}^{n - 1} } \right)}}{ \left \{ \dfrac{ \sin \left( {2}^{n} \cdot \dfrac{x}{ {2}^{n} }\right) }{ {2}^{n} \cdot\sin \left( \dfrac{x}{ {2}^{n} }\right)} \right \}^{2} }\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot \lim_{n \to \infty}\dfrac{ \dfrac{\sin( 2x )}{ {2}^{n} \cdot\sin \left( \dfrac{x}{ {2}^{n - 1} } \right)}}{ \dfrac{ \sin^{2} (x ) }{ {2}^{2n} \cdot\sin^{2} \left( \dfrac{x}{ {2}^{n} }\right)}}\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot \lim_{n \to \infty}\dfrac{ {2}^{2n} \cdot\sin^{2} \left( \dfrac{x}{ {2}^{n} }\right) \cdot\sin( 2x )}{ {2}^{n} \cdot\sin \left( \dfrac{x}{ {2}^{n - 1} } \right) \cdot\sin^{2} (x ) }\: dx}$

$\displaystyle \rm{ = \int \frac{1}{x} \cdot \lim_{n \to \infty}\dfrac{ {2}^{2n} \cdot\sin^{2} \left( \dfrac{x}{ {2}^{n} }\right) \cdot2\sin( x ) \cos(x) }{ {2}^{n} \cdot\sin \left( \dfrac{x}{ {2}^{n - 1} } \right) \cdot\sin^{2} (x ) }\: dx}$

$\displaystyle \rm{ = \int \lim_{n \to \infty} \dfrac{ x \cdot {2}^{2n} \cdot\sin^{2} \left( \dfrac{x}{ {2}^{n} }\right) \cdot \cos(x) }{ x^{2} \cdot{2}^{n - 1} \cdot\sin \left( \dfrac{x}{ {2}^{n - 1} } \right) \cdot\sin (x ) }\: dx}$

$\displaystyle \rm{ = \int \lim_{n \to \infty} \dfrac{ \dfrac{x}{ {2}^{ n - 1} } }{\sin \left( \dfrac{x}{ {2}^{n - 1} } \right)} \cdot\lim_{n \to \infty} \dfrac{ \dfrac{x^{2} }{ {2}^{ 2n } } }{\sin^{2} \left( \dfrac{x}{ {2}^{n } } \right)} \cdot\dfrac{ \cos(x) }{ \sin (x ) }\: dx}$

$\displaystyle \rm{ = \int 1 \cdot1 \cdot\dfrac{ \cos(x) }{ \sin (x ) }\: dx}$

$\displaystyle \rm{ = \int \cot(x) \: dx}$

$\displaystyle \rm{ = \ln | \sin(x)| + c}$