calculate-0-2pi-d-1-cos-3-

Question Number 42232 by maxmathsup by imad last updated on 20/Aug/18

c a l c u l a t e \int_{0}^{2 π} \frac{d θ}{{(1 + c o s θ)}^{3}}

Answered by MJS last updated on 21/Aug/18

simply Weierstrass

t = \tan \frac{θ}{2} \to d θ = \frac{2 t d}{t^{2} + 1}

\int \frac{d θ}{{(1 + \cos θ)}^{3}} = \frac{1}{4} \int {(t^{2} + 1)}^{2} d t =

= \frac{1}{4} \int t^{4} d t + \frac{1}{2} \int t^{2} d t + \frac{1}{4} \int d t =

= \frac{1}{20} t^{5} + \frac{1}{6} t^{3} + \frac{1}{4} t = \frac{(3 t^{4} + 10 t^{2} + 15) t}{60} =

= \frac{({3 \tan}^{4} \frac{x}{2} + {10 \tan}^{2} \frac{x}{2} + 15) \tan \frac{x}{2}}{60} + C

Facebook Tweet Pin