let-f-0-1-x-2-2-cos-x-1-dx-with-R-1-calculate-f-2-find-the-value-of-g-0-1-xsin-x-2-2cos-x-1-dx-

Question Number 54936 by maxmathsup by imad last updated on 14/Feb/19

l e t f (θ) = \int_{0}^{1} \sqrt{x^{2} + 2 (c o s θ) x + 1} d x w i t h θ \in R .

1) c a l c u l a t e f (θ)

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

Commented by Abdo msup. last updated on 16/Feb/19

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

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

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

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

l e t f i n d \int \sqrt{1 + t^{2}} d t c h a n g e m e n t t = s h (u) g i v e

\int \sqrt{1 + t^{2}} d t = \int c h (u) c h (u) d u

= \int {c h}^{2} (u) d u = \int \frac{1 + c h (2 u)}{2} d u

= \frac{u}{2} + \frac{1}{4} s h (2 u) = \frac{u}{2} + \frac{1}{2} c h (u) s h (u)

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

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

\int_{c o t a n (θ)}^{c o t a n (\frac{θ}{2})} \sqrt{1 + t^{2}} d t = \frac{1}{2} {[l n (t + \sqrt{1 + t^{2}}) + t \sqrt{1 + t^{2}}]}_{c o t a n θ}^{c o t a n (\frac{θ}{2})}

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

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

- c o t a n θ \sqrt{1 + {c o t a n}^{2} θ})

Answered by tanmay.chaudhury50@gmail.com last updated on 15/Feb/19

\int \sqrt{x^{2} + 2 x c o s θ + 1} d x

\int \sqrt{{(x + c o s θ)}^{2} + {s i n}^{2} θ} d x

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

s o a n s w e r i s

∣ \frac{(x + c o s θ)}{2} \sqrt{{(x + c o s θ)}^{2} + {s i n}^{2} θ} + \frac{{s i n}^{2} θ}{2} \sqrt{{(x + c o s θ)}^{2} + {s i n}^{2} θ} ∣_{0}^{1}

[{\frac{(1 + c o s θ)}{2} \sqrt{{(1 + c o s θ)}^{2} + {s i n}^{2} θ} + \frac{{s i n}^{2} θ}{2} \sqrt{{(1 + c o s θ)}^{2} + {s i n}^{2} θ}} - {\frac{c o s θ}{2} \sqrt{{c o s}^{2} θ + {s i n}^{2} θ} + \frac{{s i n}^{2} θ}{2} \times \sqrt{{c o s}^{2} θ + {s i n}^{2} θ}}]

[\frac{(1 + c o s θ)}{2} \sqrt{1 + 2 c o s θ + 1} + \frac{{s i n}^{2} θ}{2} \sqrt{2 + 2 c o s θ}} - {\frac{c o s θ}{2} + \frac{{s i n}^{2} θ}{2}}]

\frac{(1 + c o s θ)}{2} \times 2 c o s \frac{θ}{2} + \frac{{s i n}^{2} θ}{2} \times 2 c o s \frac{θ}{2} - \frac{c o s θ}{2} - \frac{{s i n}^{2} θ}{2}]

c h e c k u p t o t h i s \dots