Key to "How to find ∫−π2π2sin⁡ysin⁡y+cdy"

redupticslaz

Answered question

2022-04-22

How to find

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\sin y}{\sqrt{\sin y + c}} d y

despescarwh9

Beginner2022-04-23Added 15 answers

Using formula

\int_{φ}^{\frac{π}{2}} \frac{\sin^{m} u}{\sqrt{a + b \sin; u}} d u = \frac{2}{\sqrt{a + b}} \int_{0}^{F (\arcsin (\sqrt{\frac{1 - \sin; φ}{2}}) ∣ \frac{2 b}{a + b})} {(1 - 2 s n^{2} (u ∣ \frac{2 b}{a + b}))}^{m} d u

for

a > b > 0

, where

F (ϕ ∣ m)

is the incomplete elliptic integral of the first kind and

s n (u ∣ m)

is a Jacobian elliptic function, we have

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\sin y}{\sqrt{\sin y + c}} d y = 2 (\sqrt{1 + c}; E (\frac{2}{1 + c}) - \frac{c}{\sqrt{1 + c}} K (\frac{2}{1 + c}))

where K(m) is the complete elliptic integral of the first kind, and E(m) is the complete elliptic integral of the second kind.

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