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Question:

The integral ∫dx/(1+√x)√(1-x) is equal to (where C is a constant of integration):

−√(1−√x)/(1+√x)+C

√(1−√x)/(1+√x)+C

2√(1+√x)/(1−√x)+C

√(1+√x)/(1−√x)+C

Solution:

I=∫dx/(1+√x)√(1−x)
Put 1+√x=t ⇒ 1/(2√x)dx=dt ⇒ I=∫2dt/t√(2t−t²)
Again put t=1/z ⇒ dt=−1/z²dz ⇒ I=2∫(−1/z²)dz/(1/z)√(2/z−1/z²)=2∫−dz/√(2z−1)=−2∫dz/√(2z−1)
=−√(2z−1)+c=−√(2/t−1)+c=−√((2−t)/t)+c=−√(1−√x)/(1+√x)+c