Calculus

Integration Techniques

Integration Techniques: Level 4 Challenges

         

12x{x}{x}2x2x2dx\large \int_{1}^{\infty} \dfrac {2x\{x\} - \{x\}^2} {x^2 \lfloor x \rfloor ^2} \, dx

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

Notations:

0sinxxexdx=?\large \int_0^\infty \frac {\sin x}{xe^x} \, dx = \, ?

A=1(1u21u4)dulnu \large A = \displaystyle\int_{1}^{\infty} \left(\dfrac{1}{u^2} - \dfrac{1}{u^4}\right) \dfrac{du}{\ln u}

Find the value of eAe^A.

Let f(x)=x+ex1f(x) = x + e^x - 1, and let f1(x)f^{-1} (x) denote the inverse function of f(x)f(x) .

Then the integral e1+e2f1(x)dx \displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx evaluates to k0+k1e1+k2e2++knen, k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n , where k0,k1,,knk_0, k_1, \ldots, k_n are rational numbers and e2.718e\approx 2.718 is Euler's number.

If the sum k0+k1+k2++knk_0 + k_1 + k_2 + \cdots + k_n can be expressed as AB,\frac{A}{B}, where AA and BB are coprime positive integers, what is A+B+n?A+B+n?

The graph of f(x)=sin(lnx) f(x) = \sin ( \ln x ) (as shown above) looks innocent enough to noticeably oscillate as xx increases. However, as xx approaches 00, the oscillations grow rapidly, making f(x+ϵ) f(x + \epsilon) vary greatly from f(x) f(x) around this region, even at very infinitesimal values of ϵ \epsilon .

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

If the sum of these regions is AA, then determine 105A \big\lfloor 10^5 A \big\rfloor .

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