Question-82573

Question Number 82573 by Power last updated on 22/Feb/20

Commented by Tony Lin last updated on 22/Feb/20

l e t x = a s e c θ, d x = a s e c θ t a n θ d θ

\Rightarrow s e c θ = \frac{x}{a}, t a n θ = \frac{\sqrt{x^{2} - a^{2}}}{a}

a^{2} \int {t a n}^{2} θ s e c θ d θ

= a^{2} \int ({s e c}^{2} θ - 1) s e c θ d θ

= a^{2} [\int {s e c}^{3} θ d θ - \int s e c θ d θ]

\int {s e c}^{3} d θ

= s e c θ t a n θ - \int {t a n}^{2} θ s e c θ d θ

= s e c θ t a n θ - \int ({s e c}^{2} θ - 1) s e c θ d θ

= s e c θ t a n θ - \int {s e c}^{3} d θ + \int s e c θ d θ

\Rightarrow \int {s e c}^{3} d θ

= \frac{1}{2} (s e c θ t a n θ + l n ∣ s e c θ + t a n θ ∣) + c

∴ a^{2} \int {t a n}^{2} θ s e c θ d θ

= \frac{1}{2} a^{2} (s e c θ t a n θ - l n ∣ s e c θ + t a n θ ∣) + c

= \frac{1}{2} (x \sqrt{x^{2} - a^{2}} - a l n ∣ x + \sqrt{x^{2} - a^{2}} ∣) + c

Commented by mathmax by abdo last updated on 22/Feb/20

A = \int \sqrt{x^{2} - a^{2}} d x c h a n g e m e n t x = a c h (t) g i v e

A = \int ∣ a ∣ s h (t) a s h (t) d t = a ∣ a ∣ \int {s h}^{2} t d t

= a ∣ a ∣ \int \frac{c h (2 t) - 1}{2} = \frac{a ∣ a ∣}{2} \int c h (2 t) d t - \frac{a ∣ a ∣ t}{2}

= \frac{a ∣ a ∣}{4} s h (2 t) - \frac{a ∣ a ∣}{2} t + c = \frac{a ∣ a ∣}{2} s h (t) c h (t) - \frac{a ∣ a ∣}{2} t + c

= \frac{a ∣ a ∣}{2} \sqrt{\frac{x^{2}}{a^{2}} - 1} \times \frac{x}{a} - \frac{a ∣ a ∣}{2} a r g c h (\frac{x}{a}) + C

= \frac{a ∣ a ∣}{2 ∣ a ∣ a} x \sqrt{x^{2} - a^{2}} - \frac{a ∣ a ∣}{2} l n (\frac{x}{a} + \sqrt{\frac{x^{2}}{a^{2}} - 1}) + C

= \frac{x}{2} \sqrt{x^{2} - a^{2}} - \frac{a ∣ a ∣}{2} l n (\frac{x}{a} + \frac{\sqrt{x^{2} - a^{2}}}{∣ a ∣}) + C

Commented by Power last updated on 22/Feb/20

thanks

Commented by msup trace by abdo last updated on 22/Feb/20

y o u a r e w e l c o m e