let-a-2-gt-b-2-c-2-calculate-0-2pi-d-a-bsin-c-cos-

Question Number 46850 by maxmathsup by imad last updated on 01/Nov/18

l e t a^{2} > b^{2} + c^{2} c a l c u l a t e \int_{0}^{2 π} \frac{d θ}{a + b s i n θ + c c o s θ}

Commented by maxmathsup by imad last updated on 01/Nov/18

r e s i d u s m e t h o d l e t I = \int_{0}^{2 π} \frac{d θ}{a + b s i n θ + c c o s θ} c h a n g e m e n t z = e^{i θ} g i v e

I = \int_{∣ z ∣= 1} \frac{1}{a + b \frac{z - z^{- 1}}{2 i} + c \frac{z + z^{- 1}}{2}} \frac{d z}{i z} = \int_{∣ z ∣= 1} \frac{d z}{i z {a + b \frac{z - z^{- 1}}{2 i} + c \frac{z + z^{- 1}}{2}}}

= \int_{∣ z ∣= 1} \frac{2 d z}{z {2 i a + b (z - z^{- 1}) + c i (z + z^{- 1})}}

= \int_{∣ z ∣= 1} \frac{2 d z}{2 i a z + {b z}^{2} - b + {c i z}^{2} + c i} = \int_{∣ z ∣= 1} \frac{2 d z}{(b + c i) z^{2} + 2 i a z + c i - b}

l e t φ (z) = \frac{2}{(b + c i) z^{2} + 2 i a z + c i - b} . p o l e s o f φ ?

Δ^{^{'}} = {(i a)}^{2} - (b + c i) (c i - b) = - a^{2} - ({(c i)}^{2} - b^{2})

= - a^{2} + b^{2} + c^{2} = - (a^{2} - (b^{2} + c^{2})) < 0 \Rightarrow z_{1} = \frac{- i a + i \sqrt{a^{2} - b^{2} - c^{2}}}{b + c i}

z_{2} = \frac{- i a - i \sqrt{a^{2} - b^{2} - c^{2}}}{b + c i}

∣ z_{1} ∣ - 1 = \frac{∣ a - \sqrt{a^{2} - b^{2} - c^{2}} ∣}{\sqrt{b^{2} + c^{2}}} > 1 \Rightarrow z_{1} i s o u t o f c i r c l e . \Rightarrow R e s (φ, z_{1}) = 0

∣ z_{2} ∣ - 1 = \frac{∣ a + \sqrt{a^{2} - b^{2} - c^{2}} ∣}{\sqrt{b^{2} + c^{2}}} > 1 \Rightarrow z_{2} i s o u t o f c i r c l e \Rightarrow R e s (φ, z_{2}) = 0

\int_{0}^{2 π} φ (z) d z = 0

Answered by tanmay.chaudhury50@gmail.com last updated on 01/Nov/18

t = t a n \frac{θ}{2} d t = {s e c}^{2} \frac{θ}{2} \times \frac{1}{2} d θ

d θ = \frac{2 d t}{1 + t^{2}}

\int \frac{2 d t}{(1 + t^{2}) (a + b \times \frac{2 t}{1 + t^{2}} + c \times \frac{1 - t^{2}}{1 + t^{2}})}

\int \frac{2 d t}{(a + {a t}^{2} + 2 b t + c - {c t}^{2})}

2 \int \frac{d t}{t^{2} (a - c) + t (2 b) + a + c}

\frac{2}{a - c} \int \frac{d t}{t^{2} + 2 . t . \frac{b}{a - c} + \frac{a + c}{a - c}}

\frac{2}{a - c} \int \frac{d t}{{(t + \frac{b}{a - c})}^{2} + \frac{a + c}{a - c} - \frac{b^{2}}{{(a - c)}^{2}}}

\frac{2}{a - c} \int \frac{d t}{{(t + \frac{b}{a - c})}^{2} + \frac{a^{2} - c^{2} - b^{2}}{{(a - c)}^{2}}}

\frac{2}{a - c} \times \frac{1}{\sqrt{\frac{a^{2} - b^{2} - c^{2}}{{(a - c)}^{2}}}} \times {t a n}^{- 1} (\frac{t + \frac{b}{a - c}}{\sqrt{\frac{a^{2} - b^{2} - c 2}{{(a - c)}^{2}}}})

= \frac{2}{\sqrt{a^{2} - b^{2} - c^{2}}} {t a n}^{- 1} (\frac{t + \frac{b}{a - c}}{\sqrt{\frac{a^{2} - b^{2} - c^{2}}{{(a - c)}^{2}}}})

w h e n θ \to 0 t \to 0

θ \to 2 π t \to 0

t h u s v a l u e o f i n y r e g a t i o n z e r o

p l s c h e c k w h e r e i a m w r o n g . .

s i n c e a^{2} > (b^{2} + c^{2})