Rationalization Method to Remove Indeterminate Form
If ∫dxx-4√x+5...
Question
If ∫dx(x−4)√x+5=13ln∣∣
∣∣√f(x)−α√f(x)+β∣∣
∣∣+C where (α,β,C) are constants, then which of the following is/are CORRECT?
A
α=β
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B
α≠β
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C
f(x)=x+5
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D
f(x)=√x+5
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Solution
The correct option is Cf(x)=x+5 Let, I=∫dx(x−4)√x+5
Put x+5=t2 ⇒dx=2tdt ⇒I=∫2tdt(t2−9)(t)=∫2dtt2−9{∵∫1t2−a2dx=12aln∣∣∣t−at+a∣∣∣+C}=13ln∣∣∣t−3t+3∣∣∣+C=13ln∣∣∣√x+5−3√x+5+3∣∣∣+C