-
Looking for HSC notes and resources? Check out our Notes & Resources page
Integration (1 Viewer)
- Thread starter Arowana21
- Start date
- Joined
- Feb 16, 2005
- Messages
- 8,132
- Gender
- Male
- HSC
- 2006
Note that:
x4/(1 - x) = - {(1 - x4) - 1} / (1 - x)
= - (1 - x4) / (1 - x) + 1 / (1 - x)
Note the first term (ignoring the minus in front) is a geometric series formula for 1 + x + x² + x³: [alternatively you can use 1 - x4 = (1 - x)(1 + x)(1 + x²)]
= - (1 + x + x² + x³) + 1 / (1 - x)
∫[x4/(1 - x)] dx = - ∫(1 + x + x² + x³)dx + ∫[dx / (1 - x)]
= - (x + x²/2 + x³/3 + x4/4) - ln (1 - x) + c
x4/(1 - x) = - {(1 - x4) - 1} / (1 - x)
= - (1 - x4) / (1 - x) + 1 / (1 - x)
Note the first term (ignoring the minus in front) is a geometric series formula for 1 + x + x² + x³: [alternatively you can use 1 - x4 = (1 - x)(1 + x)(1 + x²)]
= - (1 + x + x² + x³) + 1 / (1 - x)
∫[x4/(1 - x)] dx = - ∫(1 + x + x² + x³)dx + ∫[dx / (1 - x)]
= - (x + x²/2 + x³/3 + x4/4) - ln (1 - x) + c