Evaluate the following Integral: ∫ x(x - 4)^1/3 dx, x ∈ [4, 12]

Question

Evaluate the following Integral:

\(\int\limits_4^{12} \)x(x - 4)^\(\frac13\)dx

Answer 1

Let I =  \(\int\limits_4^{12} \)x(x - 4)^\(\frac13\)dx

Put x – 4 = t³

⇒ dx = 3t²dt (Differentiating both sides)

When x = 4, t³ = 4 – 4 = 0 ⇒ t = 0

When x = 12, t³ = 12 – 4 = 8 ⇒ t = 2

So, the new limits are 0 and 2.

We can write x = t³ + 4

Substituting this in the original integral,