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प्रश्न
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
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उत्तर
`int ("d"x)/sqrt(x^2 - 4x + 5) = int ("d"x)/sqrt((x - 2)^2 - 2^2 + 5)`
= `int ("d"x)/sqrt((x - 2)^2 - 4 + 5)`
= `int ("d"x)/sqrt((x - 2)^2 + 1)`
Put x – 2 = t
dx = dt
`int ("d"x)/sqrt(x^2 - 4x + 5) = int ("d"x)/sqrt("t"^2 + 1^2)`
`int ("d"x)/sqrt(x^2 + "a"^2) = log |x + sqrt(x^2 + "a"^2)| + "c"`
= `log|"t" + sqrt("t"^2 + 1^2)| + "c"`
= `log|(x - 2) + sqrt((x - 2)^2 + 1)| + "c"`
= `log|(x - 2) + sqrt(x^2 - 4x + 4x +1)| + "c"`
`int ("d"x)/sqrt(x^2 - 4x + 5) = log |(x - 2) + sqrt(x^2 - 4x + 5)| + "c"`
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