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Question
Evaluate the following:
`int sqrt(2"a"x - x^2) "d"x`
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Solution
Let I = `int sqrt(2"a"x - x^2) "d"x`
= `int sqrt(-(x^2 - 2"a"x)) "d"x`
= `int sqrt(-(x^2 - 2"a"x + "a"^2 - "a"^2)) "d"x`
= `int sqrt(-[(x - "a")^2 - "a"^2]) "d"x`
= `int sqrt("a"^2 - (x - "a")^2) "d"x`
= `(x - "a")/2 sqrt("a"^2 - x^2) + "a"^2/2 sin^-1 ((x - "a")/"a") + "C"` ......`[because int sqrt("a"^2 - x^2) "d"x = x/2sqrt("a"^2 - x^2) - "a"^2/2 sin^-1 x/"a" + "C"]`
= `(x - "a")/2 sqrt("a"^2 - (x^2 - 2"a"x + "a"^2)) + "a"^2/2 sin^-1 ((x - "a")/"a") + "C"`
= `(x - "a")/2 sqrt(2"a"x - x^2) + "a"^2/2 sin^-1 9(x - "a"0/"a") + "C"`
Hence, I = `(x - "a")/2 sqrt(2"a"x - x^2) + "a"^2/2 sin^-1 ((x - "a")/"a") + "C"`.
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