Evaluate the following Integral: ∫(24 x^3)/(1 + x^2)^4dx, x ∈ [0, 1]

Question

Evaluate the following Integral:

\(\int\limits_0^1\cfrac{24\text x^3}{(1+\text x^2)^4}d\text x \)

Answer 1

Let I = \(\int\limits_0^1\cfrac{24\text x^3}{(1+\text x^2)^4}d\text x \)

Put 1 + x 2 = t

⇒ 2xdx = dt (Differentiating both sides)

When x = 0, t = 1 + 0² = 1

When x = 1, t = 1 + 1² = 2

So, the new limits are 1 and 2.

In numerator, we can write 24x³dx = 12x² × 2xdx

But, x² = t – 1 and 2xdx = dt

⇒ 24x³dx = 12(t – 1)dt

Substituting this in the original integral,