Question

Integral logaritma natural
mohon bantuannya

Answer 1

Anggap:
p = x² + 4 maka dp = 2x dx
p - 4 = x²
(p - 4)² = x^4
maka
integral ln(x² + 4) / x³ dx
= integral x ln (x² + 4) / x^4 dx
= integral ln (p) / (p - 4)² dp/2
= integral ln (p) (p - 4)^-2 dp/2

Pake parsial
anggap:
u = ln p maka du = 1/p dp
dv = (p - 4)^-2 dp
v = -(p - 4)^-1
integral ln (p) (p - 4)^-2 dp/2
= 1/2 . [ln p . -(p - 4)^-1 - Integral -(p - 4)^-1 . 1/p dp
= 1/2 . [-ln p / (p - 4) + Integral 1/p(p - 4) dp]

Menentukan:
Integral 1/p(p - 4) dp
= Integral 1/4(p - 4) - 1/4p dp
= 1/4 [ln (p-4) - ln p]

maka
integral ln (p) (p - 4)^-2 dp/2
= 1/2 . [-ln p / (p - 4) + Integral 1/p(p - 4) dp]
= 1/2 . [-ln p / (p - 4) + 1/4 [ln (p-4) - ln p]]
= 1/8 . [-4 ln p / (p - 4) + ln (p-4) - ln p]
= 1/8 . [-4 ln (x² + 4) / (x² + 4 - 4) + ln (x² + 4 - 4) - ln (x² + 4)]
= 1/8 . [-4 ln (x² + 4) / x² + ln (x²) - ln (x² + 4)]
= 1/8x² . [-4 ln (x² + 4) + x² ln (x²) - x² ln (x² + 4)]
= 1/8x² . [x² ln (x²) - (x²+4) ln (x² + 4)]
= 1/8x² . [2x² ln (x) - (x²+4) ln (x² + 4)] + c