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
$$\int_0^\infty \frac{\arctan{\pi x}-\arctan{x}}{x}dx=F(\pi)=\frac{\pi}{2}\ln{\pi}$$
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
$$\int_0^\infty \frac{\arctan{\pi x}-\arctan{x}}{x}dx=F(\pi)=\frac{\pi}{2}\ln{\pi}$$
{ 0 Tanggapan pembaca... read them below or add one }
Posting Komentar
Tanggapan Anda sangat berarti bagi kami