0-tan-a-2-cos-2-b-2-sin-2-d-

Question Number 35949 by ajfour last updated on 26/May/18

\int_{0}^{α} \frac{\tan θ}{\sqrt{a^{2} \cos^{2} θ - b^{2} \sin^{2} θ}} d θ = ?

Commented by abdo mathsup 649 cc last updated on 26/May/18

l e t p u t I_{α} = \int_{0}^{α} \frac{t a n θ}{\sqrt{a^{2} {c o s}^{2} θ - b^{2} {s i n}^{2} θ}} d θ

w e h s v e t h e f o r m u l a 1 + {t a n}^{2} θ = \frac{1}{{c o s}^{2} θ} \Rightarrow

{c o s}^{2} θ = \frac{1}{1 + {t a n}^{2} θ} a n d {s i n}^{2} θ = 1 - \frac{1}{1 + {t a n}^{2} θ} \Rightarrow

a^{2} {c o s}^{2} θ - b^{2} {s i n}^{2} θ

= \frac{a^{2}}{1 + {t a n}^{2} θ} - b^{2} \frac{{t a n}^{2} θ}{1 + {t a n}^{2} θ} = \frac{a^{2} - b^{2} {t a n}^{2} θ}{1 + {t a n}^{2} θ} \Rightarrow

I_{α} = \int_{0}^{α} \frac{t a n θ \sqrt{1 + {t a n}^{2} θ}}{\sqrt{a^{2} - b^{2} {t a n}^{2} θ}} d θ c h a n g e m e n t t a n θ = t

g i v e I_{α} = \int_{0}^{t a n α} \frac{t \sqrt{1 + t^{2}}}{\sqrt{a^{2} - b^{2} t^{2}}} \frac{d t}{1 + t^{2}}

= \int_{0}^{t a n (α)} \frac{t}{\sqrt{1 + t^{2}}} \frac{d t}{\sqrt{a^{2} - b^{2} t^{2}}} b y p a r t s

u^{^{'}} = \frac{t}{\sqrt{1 + t^{2}}} a n d v = \frac{1}{\sqrt{a^{2} - b^{2} t^{2}}}

I_{α} = {[\frac{\sqrt{1 + t^{2}}}{\sqrt{a^{2} - b^{2} t^{2}}}]}_{0}^{t a n (α)} - \int_{0}^{t a n (α)} \sqrt{1 + t^{2}} (- \frac{1}{2} - 2 b^{2} t) {(a^{2} - b^{2} t^{2})}^{- \frac{3}{2}} d t

= \frac{\sqrt{1 + {t a n}^{2} α}}{\sqrt{a^{2} - b^{2} {t a n}^{2} α}} - \frac{1}{∣ a ∣} - b^{2} \int_{0}^{t a n (α)} \sqrt{1 + t^{2}} {(a^{2} - b^{2} t^{2})}^{- \frac{3}{2}} d t

l e t J = \int_{0}^{t a n (α)} \sqrt{1 + t^{2}} {(a^{2} - b^{2} t^{2})}^{- \frac{3}{2}} d t

= c \int_{0}^{t a n (α)} \frac{\sqrt{1 + t^{2}}}{(a^{2} - b^{2} t^{2}) \sqrt{a^{2} - b^{2} t^{2}}} d t \dots b e c o n t i n u e d \dots

Answered by tanmay.chaudhury50@gmail.com last updated on 26/May/18

\int_{0}^{α} \frac{t a n θ . s e c θ}{\sqrt{a^{2} - b^{2} {t a n}^{2} θ}} d θ

\int_{0}^{α} \frac{t a n θ s e c θ}{\sqrt{a^{2} - b^{2} {s e c}^{2} θ + b^{2}}} d θ

\frac{1}{b} \int \frac{d (s e c θ)}{\sqrt{\frac{a^{2} + b^{2}}{b^{2}} - {s e c}^{2} θ}}

n o w u s e f o r m u l a \int \frac{d x}{\sqrt{A^{2} - X^{2}}}

\frac{1}{b} \times ∣ {s i n}^{- 1} (\frac{s e c θ}{\sqrt{\frac{a^{2} + b^{2}}{b^{2}}}}) ∣_{0}^{α}

\frac{1}{b} \times {{s i n}^{- 1} (\frac{s e c α}{\sqrt{\frac{a^{2} + b^{2}}{b^{2}}}}) - {s i n}^{- 1} (\frac{1}{\frac{\sqrt{a^{2} + b^{2}}}{b^{2}}})}

Commented by ajfour last updated on 26/May/18

T h a n k s .

Commented by tanmay.chaudhury50@gmail.com last updated on 26/May/18

i t s o k \dots

Commented by ajfour last updated on 26/May/18

i a p p l i e d y o u r t e c h n i q u e a g a i n

a n d c o r r e c t e d m y a n s w e r t o

Q . 35940 . T h a n k s a g a i n .

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