Eduprobe
Question:
Evaluate ∫dx/[x(x³+1)]
Solution:
Let I = ∫dx/[x(x³+1)] ⇒ (1/3)∫3x²dx/[x³(x³+1)]
Put x³+1 = t ⇒ 3x²dx = dt
i.e., I = (1/3)∫dt/[t(t-1)]
= (1/3)∫[1/(t-1) - 1/t]dt = (1/3)[log|t-1| - log|t|] + C
I = (1/3)log|x³/(x³+1)| + C
Let I = ∫dx/[x(x³+1)] ⇒ (1/3)∫3x²dx/[x³(x³+1)]
Put x³+1 = t ⇒ 3x²dx = dt
i.e., I = (1/3)∫dt/[t(t-1)]
= (1/3)∫[1/(t-1) - 1/t]dt = (1/3)[log|t-1| - log|t|] + C
I = (1/3)log|x³/(x³+1)| + C