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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.

Level 4

$\large \int_{0}^{\infty} \dfrac{\tan^{-1}(\pi x) - \tan^{-1}(x)}{x}\, dx = A\pi^{B}\dfrac{\ln \pi}{C}$

The equation holds true for positive integers $$A,B,C$$ with $$A,C$$ coprime. Find $$A\times B\times C$$.

$\large I = \int_{0}^{\infty}\dfrac{x \ln{(4x)}}{\frac{1}{4}x^{4}+x^{2}+1} \,dx$

$$I$$ is the value of the closed form of the above integral. What is the value of $$\lfloor 100I \rfloor$$?

You may use the fact that $$\ln(2) \approx 0.693147$$.

$\large \displaystyle \int_0^\pi \dfrac{x\sin 2x \sin\left(\frac \pi 2 \cos x\right)}{2x-\pi} \, dx$

The above integral can be expressed as $$\dfrac{A}{B\pi^C}$$ where $$A$$, $$B$$ and $$C$$ are positive integers with $$A$$, $$B$$ coprime. Find $$A+B+C$$

$\large \int_{\frac{1}{8}}^{\frac{1}{2}} \left \lfloor \ln \left \lceil \dfrac{1}{x} \right \rceil \right \rfloor dx = \, ?$

 Notations: $$\lfloor \cdot \rfloor$$ denotes the floor function and $$\lceil \cdot \rceil$$ denotes the ceiling function.

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

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