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Question
Evaluate the following:
`int sqrt(("a" + x)/("a" - x)) "d"x`
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Solution
Let I = `int sqrt(("a" + x)/("a" - x)) "d"x`
Put x = `"a" cos 2theta`
⇒ dx = `-"a" * sin 2theta * 2 * "d"theta`
∴ I = `-2int sqrt(("a" + "a" cos 2theta)/("a" - "a" cos 2theta)) * "a" sin 2theta "d"theta`
= `-2"a" int sqrt((1 + cos 2theta)/(1 - cos 2theta)) sin 2theta "d"theta`
= `-2"a" int sqrt((2 cos^2theta)/(2 sin^2 theta)) sin 2theta "d"theta`
= `-2"a" int cot theta * sin 2theta "d"theta`
= `-2"a" int costheta/sintheta * 2 sin theta cos theta "d" theta`
= `-4"a" int cos^2theta "d"theta`
= `-2"a" int (1 + cos 2theta)"d"theta`
= `-2"a" [theta + 1/2 sin 2theta] + "C"`
= `-2"a" [1/2 cos^-1 x/"a" + 1/2 sqrt(1 - x^2/"a"^2)] + "C"`
= `-"a" [cos^-1(x/"a") + sqrt(1 - x^2/"a"^2)] + "C"`
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