Q.163. Definite Integral

Question 163.
Using the substitution x = sec^2 t, find  ∫ dx / [x^2 * √(x - 1)] from x = 1 to x = 2.

Answer 163.
x = sec^2 t => dx = 2sec^2 t tant
=> Integration
= ∫ 2sec^2 t tant dt / [sec^4 t (sec^2 t - 1)^(1/2)]
= ∫ 2cos^2 t dt
= ∫(1 + cos2t) dt
= t + (1/2) sin2t + c
= t + sint cost + c
= t + tant / sec^2 t + c
= arcsec(√x) + √(x - 1) / x + c
Plugging limits, 1 to 2,
= arcsec(√2) + 1/2 - (arcsec1)
= π/4 + 1/2 - 0
= π/4 + 1/2.

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