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Integration Techniques

Writing an integral down is only the first step. A toolkit of techniques can help find its value, from substitutions to trigonometry to partial fractions to differentiation.

Integration Techniques: Level 4 Challenges

         

\[ \displaystyle \int_{1}^{\infty} \dfrac {2x\{x\} - \{x\}^2} {x^2 \lfloor x \rfloor ^2} \, dx \]

If the integral above can be expressed as \( \dfrac{\pi^a - b} c \), where \(a,b,c\) are all positive integers, find \(a+b+c\).

Notations:

\[\large \int_0^\infty \frac {\sin x}{xe^x} \, dx = \, ? \]

\[ \large A = \displaystyle\int_{1}^{\infty} \left(\dfrac{1}{u^2} - \dfrac{1}{u^4}\right) \dfrac{du}{\ln u} \]

Find the value of \(e^A\).

Let \(f(x) = x + e^x - 1\), and let \(f^{-1} (x) \) denote the inverse function of \(f(x) \).

Then the integral \( \displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx \) evaluates to \[ k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n ,\] where \(k_0, k_1, \ldots, k_n\) are rational numbers and \(e\approx 2.718\) is Euler's number.

If the sum \(k_0 + k_1 + k_2 + \cdots + k_n\) can be expressed as \(\frac{A}{B},\) where \(A\) and \(B\) are coprime positive integers, what is \(A+B+n?\)

The graph of \( f(x) = \sin ( \ln x ) \) (as shown above) looks innocent enough to noticeably oscillate as \(x\) increases. However, as \(x\) approaches \(0\), the oscillations grow rapidly, making \( f(x + \epsilon) \) vary greatly from \( f(x) \) around this region, even at very infinitesimal values of \( \epsilon \).

That said, \( f(x) \) will cross the \(x\)-axis for an infinite number of times from \(x=0\) to \(x=1\), creating several regions of the first quadrant enclosed by the curve and the \(x\)-axis.

If the sum of these regions is \(A\), then determine \( \big\lfloor 10^5 A \big\rfloor \).

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