1-let-0-lt-lt-pi-2-and-A-0-pi-2-dx-x-2-2sin-x-1-calculate-A-2-calculate-0-pi-2-dx-x-2-2-x-1-

Question Number 53463 by maxmathsup by imad last updated on 22/Jan/19

1) l e t 0 < θ < \frac{π}{2} a n d A (θ) = \int_{0}^{\frac{π}{2}} \frac{d x}{\sqrt{x^{2} + 2 s i n θ x + 1}}

c a l c u l a t e A (θ)

2) c a l c u l a t e \int_{0}^{\frac{π}{2}} \frac{d x}{\sqrt{x^{2} + \sqrt{2} x + 1}}

Commented by maxmathsup by imad last updated on 23/Jan/19

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

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

A (θ) = \int_{t a n θ}^{\frac{π}{2 c o s θ} + t a n θ} \frac{c o s θ d t}{c o s θ \sqrt{1 + t^{2}}} = \int_{t a n θ}^{\frac{π}{2 c o s θ} + t a n θ} \frac{d t}{\sqrt{1 + t^{2}}} = {[a r g s h (t)]}_{t a n θ}^{\frac{π}{2 c o s θ} + t a n θ}

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

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

2) \int_{0}^{\frac{π}{2}} \frac{d x}{\sqrt{x^{2} + \sqrt{2} x + 1}} = A (\frac{π}{4}) = l n (\frac{π}{2 c o s (\frac{π}{4})} + t a n (\frac{π}{4}) + \sqrt{1 + (\frac{π}{2 c o s (\frac{π}{4})} + t a n {(\frac{π}{4})}^{2}}

- l n (t a n (\frac{π}{4}) + \sqrt{1 + {t a n}^{2} (\frac{π}{4})})

= l n (\frac{π}{\sqrt{2}} + 1 + \sqrt{1 + {(\frac{π}{\sqrt{2}} + 1)}^{2}}) - l n (1 + \sqrt{2}) .

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

1) \int_{0}^{\frac{π}{2}} \frac{d x}{\sqrt{x^{2} + 2 x s i n θ + 1}}

\int_{0}^{\frac{π}{2}} \frac{d x}{\sqrt{{(x + s i n θ)}^{2} + {c o s}^{2} θ}} =∣ l n {∣ (x + s i n θ) + \sqrt{{(x + s i n θ)}^{2} + {c o s}^{2} θ}) ∣_{0}^{\frac{π}{2}}

= [l n {(\frac{π}{2} + s i n θ) + \sqrt{{(\frac{π}{2} + s i n θ)}^{2} + {c o s}^{2} θ}} - l n {(0 + s i n θ) + \sqrt{{(0 + s i n θ)}^{2} + {c o s}^{2} θ}}

= l n {(\frac{π}{2} + s i n θ) + \sqrt{\frac{π^{2}}{4} + π s i n θ + 1}} - l n {s i n θ + 1}

u s i n g f o r m u l a

\int \frac{d x}{\sqrt{x^{2} + a^{2}}} = l n (x + \sqrt{x^{2} + a^{2}})

2) p u t θ = \frac{π}{4} f o r a n s w e r \dots

l n {(\frac{π}{4} + \frac{1}{\sqrt{2}}) + \sqrt{\frac{π^{2}}{4} + \frac{π}{\sqrt{2}} + 1}} - l n {\frac{1}{\sqrt{2}} + 1}

s i r p l s c h e c k \dots