Question

Evaluate the following integrals – ∫(x-2)√(2x²-6x+5)dx

\(\int(x-2)\sqrt{2x^2-6x+5}\) dx

Answer 1

Let I = ∫(x-2)√(2x²-6x+5)dx

Let us assume,

We know,

\(\frac{d}{dx}\)(xⁿ) = nx^n-1

And derivative of a constant is 0.

⇒ x – 2 = λ(2 × 2x^2-1 – 6 – 0) + μ 

⇒ x – 2 = λ(4x – 6) + μ 

⇒ x – 2 = 4λx + μ – 6λ

Comparing the coefficient of x on both sides, we get 

4λ = 1 

⇒ λ = \(\frac{1}{4}\)

Comparing the constant on both sides, we get 

μ – 6λ = –2

Substituting this value in I, we can write the integral as

Now, 

Put 2x² – 6x + 5 = t 

⇒ (4x – 6)dx = dt 

(Differentiating both sides) 

Substituting this value in I₁, we can write

Hence, we can write I₂ as