Integrate: cos(t)/(sqr root: 1 + sin^2(t)) dt?

Question

please show work. the upperlimit is pi/2 and the lowerlimit is 0.
thanks

Answer 1

Integrate from 0 to π/2.

∫[cost / √(1 + sin²t)] dt
Let
sinhθ = sint
coshθ dθ = cost dt
θ = arcsinh(sint)

= ∫[coshθ / √(1 + sinh²θ)] dθ

= ∫[coshθ / √(cosh²θ)] dθ

= ∫[coshθ / cosh²θ] dθ

= ∫dθ = θ = arcsinh(sint) | [Eval from 0 to π/2]

arcsinh(sin π/2) - arcsinh(sin 0)

= arcsinh(1) - arcsinh(0)

= ln(√2 + 1) - 0

= ln(√2 + 1)
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