Definite Integration
Integration
Grade Class 12
Question:
<span>∫</span><span><span>dx</span><span>(x-α)√((x-α)(x-β))</span></span><span>equals</span>
<span><span>2</span><span>α+β</span></span>√<span><span>x+β</span><span>x-α</span></span>+C
<span><span>-2</span><span>α-β</span></span>√<span><span>x-β</span><span>x-α</span></span>+C
<span><span>-2</span><span>α-β</span></span>√<span><span>x+β</span><span>x-α</span></span>+C
<span><span>2</span><span>α+β</span></span>√<span><span>x-β</span><span>x-α</span></span>+C
Step-by-Step Solution
Key Concept: Substitute x - alpha = 1/t to transform the integral into a standard form.
Let x - α = 1/t, then dx = -1/t^2 dt. The integral becomes ∫ (-1/t^2) / ((1/t) * √((1/t)(1/t + α - β))) dt = ∫ -1 / (t * √((1 + t(α - β))/t^2)) dt = ∫ -1 / √(1 + t(α - β)) dt. Integrating this gives the result.
Correct Answer: 2