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Problem - Integral Arctan | Catatan Nurkholis

Problem – Integral Arctan

Problem:
Hitunglah
$$int_0^infty frac{arctan{pi x}-arctan{x}}{x}dx$$

Solusi:
Misalkan $displaystyle F(a)=int_0^infty frac{arctan{a x}-arctan{x}}{x}dx$, dan $F(1)=0$. Maka $$F'(a)=int_0^infty frac{dx}{1+(ax)^2}dx=frac{pi}{2a}$$
Dengan demikian,
$$F(a)=int frac{pi}{2a}da=frac{pi}{2}ln{a}+C$$
Karena $F(1)=0$, maka $C=0$, sehingga
$$int_0^infty frac{arctan{pi x}-arctan{x}}{x}dx=F(pi)=frac{pi}{2}ln{pi}$$

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