1-calculate-f-0-1-t-2-2sin-t-1-dt-with-0-pi-2-2-calculate-g-t-0-1-t-2-2-sin-t-1-d-3-find-also-h-0-1-t-t-2-2-sin-t-1-dt-

Question Number 57666 by maxmathsup by imad last updated on 09/Apr/19

1) c a l c u l a t e f (θ) = \int_{0}^{1} \sqrt{t^{2} + 2 s i n θ t + 1} d t w i t h 0 ⩽ θ ⩽ \frac{π}{2}

2) c a l c u l a t e g (t) = \int_{0}^{1} \sqrt{t^{2} + 2 (s i n θ) t + 1} d θ

3) f i n d a l s o h (θ) = \int_{0}^{1} \frac{t}{\sqrt{t^{2} + 2 (s i n θ) t + 1}} d t

Commented by kaivan.ahmadi last updated on 10/Apr/19

t h a n k s i r, i t w a s f a l s

Commented by kaivan.ahmadi last updated on 10/Apr/19

b u t i t s e q u a l t o

{(t + s i n θ)}^{2} + 1 - {s i n}^{2} θ = {(t + s i n θ)}^{2} + {c o s}^{2} θ

Commented by maxmathsup by imad last updated on 10/Apr/19

1) w e h a v e f (θ) = \int_{0}^{1} \sqrt{t^{2} + 2 s i n θ t + {s i n}^{2} θ + {c o s}^{2}} d t

= \int_{0}^{1} \sqrt{{(t + s i n θ)}^{2} + {c o s}^{2} θ} d t c h a n g e m e n t t + s i n θ = c o s θ u g i v e

f (θ) = \int_{t a n θ}^{\frac{1 + s i n θ}{c o s θ}} c o s θ \sqrt{1 + u^{2}} c o s θ d u = {c o s}^{2} θ \int_{t a n θ}^{\frac{1 + s i n θ}{c o s θ}} \sqrt{1 + u^{2}} d u l e t f i n d

I = \int \sqrt{1 + u^{2}} d y c h a n g . u = s h α g i v e I = \int c h α c h (α) d α = \int \frac{1 + c h (2 α)}{2} d α

= \frac{1}{2} α + \frac{1}{4} s h (2 α) + c = \frac{α}{2} + \frac{1}{2} s h (α) c h (α) + c = \frac{a r g s h (u)}{2} + \frac{1}{2} u \sqrt{1 + u^{2}}

= \frac{1}{2} l n (u + \sqrt{1 + u^{2}}) + \frac{u}{2} \sqrt{1 + u^{2}} + c \Rightarrow

f (θ) = \frac{{c o s}^{2} (θ)}{2} {[l n (u + \sqrt{1 + u^{2}}) + u \sqrt{1 + u^{2}}]}_{t a n θ}^{\frac{1 + s i n θ}{c o s θ}}

= \frac{1}{2} {c o s}^{2} θ {l n (\frac{1 + s i n θ}{c o s θ} + \sqrt{1 + {(\frac{1 + c o s θ}{s i n θ})}^{2}}) + \frac{1 + s i n θ}{c o s θ} \sqrt{1 + {(\frac{1 + s i n θ}{c o s θ})}^{2}}

- l n (t a n θ + \sqrt{1 + {t a n}^{2} θ}) - t a n θ \sqrt{1 + {t a n}^{2} θ}} .