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\[ \large I_n =\int_0^1 \frac{x^n}{x^2 + a^2} \, dx \]

Define the integral \(I_n\) as above for positive real variable \(a\) independent of \(x\) and natural number ...

\[ \large I_n =\int_0^1 \frac{x^n}{ax+b} \, dx \]

Define the integral \(I_n\) as above for positive real variables \(a\) and \(b\) independent of \(x\) and natural number ...

\[ \large f(x) = \frac2{e^x - e^{-x}} \left( 1 + \int_1^x f(t) \, dt \right) \]

Suppose a function \(f\) defined on \(x>0\) satisfy the equation above, find the value ...

\[\large{\displaystyle \int^{1}_{0} \frac{\sin x(\cos^2 x-\cos^2 \frac{\pi}{5} )(\cos^2 x-\cos^2 \frac{2\pi}{5} )}{\sin 5x}.dx}\]

The ...

\[\Large \lim_{x\to6}f(f(x)).\]

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