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Question

Mathematics Question on Integrals of Some Particular Functions

∫dxx−x\int\frac{dx}{x-\sqrt{x}} is equal to

A

2 log ∣x−1∣+C2\, log\, \left|\sqrt{x}-1\right|+C

B

2 log ∣x+1∣+C2\, log\, \left|\sqrt{x}+1\right|+C

C

log ∣x−1∣+Clog\, \left|\sqrt{x}-1\right|+C

D

12log ∣x+1∣+C\frac{1}{2}log\, \left|\sqrt{x}+1\right|+C

Answer

2 log ∣x−1∣+C2\, log\, \left|\sqrt{x}-1\right|+C

Explanation

Solution

Let I=∫dxx−x=∫dxx(x−1)I=\int \frac{d x}{x-\sqrt{x}}=\int \frac{d x}{\sqrt{x}(\sqrt{x}-1)}
Put x−1=t\sqrt{x}-1=t
⇒12xdx=dt\Rightarrow \frac{1}{2 \sqrt{x}} d x=d t
∴I=∫2dtt=2log⁡∣t∣+C\therefore I=\int \frac{2 d t}{t}=2 \log |t|+C
=2log⁡∣x−1∣+C=2 \log |\sqrt{x}-1|+C