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प्रश्न
Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`
उत्तर
Let I = `int_1^oo 1/(sqrt(x)(1 + x))*dx`
Put x = tan2t
∴ dx = `[2 tan t d/dt (tan t)]*dt`
= 2 tan t sec2t·dt
When x = `oo, tan^2t = oo therefore t = pi/(2)`
When x = `1, tan^2t = 1 therefore t = pi/(4)`
∴ I = `int_(pi/4)^(pi/2) (2tantsec^2t)/(sqrt(tan^2t) (1 + tan^2t))*dt`
= `int_(pi/4)^(pi/2) (2sec^2t)/(sec^2t)*dt`
= `2 int_(pi/4)^(pi/2) 1*dt = 2[t]_(pi/4)^(pi/2)`
= `2[pi/2 - pi/4]`
= `2[pi/4]`
= `pi/(2)`.
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