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Evaluate:

\[ \large \int x \cdot \frac{\ln ( x + \sqrt{1 + x^2})}{\sqrt{1 + x^2}} dx\]

\[ \large \cos\left(\dfrac{2\pi}{13}\right)+\cos\left(\dfrac{6\pi}{13}\right)+\cos\left(\dfrac{8\pi}{13}\right)\]

If the trigonometric expression above can be expressed ...

\[ \int_3^6 \left ( \sqrt{x + \sqrt{12x-36}} + \sqrt{x - \sqrt{12x-36}} \ \right ) \ dx \]

If the definite integral above can be expressed as \(a \sqrt b\) where \(a,b\) are positive ...

\[\large{\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \ln (29+20\cos (2x)) \, dx =(A \pi^{B} \ln ( C) )^{D}}\]

Given that the above equation is true for positive ...

\[\large \int\limits_{-\infty}^{\infty} (f(x) + g(x))e^{-2\pi i \lambda x} dx = \Re - i\Im\]

where

...

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