finding integral!!!!!!!!!!!!!!!!!!!!!!!!!!

Question

Evaluate the Integral.

Answer 1

Given that,

         4∫(0 to π/4) secθtanθdθ

      = 4∫(0 to π/4) sinθ/(cosθ^2)dθ     [ Since secθ = 1/cosθ and tanθ = sinθ/cosθ ]

Let us assume that u = cosθ

Therefore derivating on both sides we get,

                                                                du/dθ = sinθ

                                                           => du = sinθdθ

If θ = 0, then u = 1 and if θ = π/4, then u = 1/√2

So 4∫(0 to π/4) secθtanθdθ

  = 4∫(1 to 1/√2) 1/u^2 du

  = -4/u (from 1 to 1/√2)

  = -4(u^-1) (from 1 to 1/√2)

  = -4(1^-1) - (-4)(1/√2)^-1

  = 4√2 - 4

  = 4(√2 - 1)

Therefore ∫(0 to π/4) secθtanθdθ = 4(√2 - 1).