0-a-1-x-a-2-x-2-dx-

Question Number 6519 by benny last updated on 30/Jun/16

\underset{0}{\int^{a}} \frac{1}{x + \sqrt{a^{2} - x^{2}}} d x =

Commented by Tawakalitu. last updated on 30/Jun/16

\frac{Π}{4}

Commented by Tawakalitu. last updated on 01/Jul/16

x = a s i n t

d x = a c o s t d t

∴ \int \frac{d x}{x + \sqrt{a^{2} - x^{2}}}

= \int \frac{a c o s t d t}{a s i n t + \sqrt{a^{2} - a^{2} {s i n}^{2} t}} = \int \frac{a c o s t d t}{a s i n t + \sqrt{a^{2} - a^{2} (1 - {c o s}^{2} t)}} = \int \frac{a c o s t d t}{a s i n t + \sqrt{a^{2} - a^{2} + a^{2} {c o s}^{2} t}}

= \int \frac{a c o s t d t}{a s i n t + \sqrt{a^{2} {c o s}^{2} t}}

= \int \frac{a c o s t d t}{a s i n t + a c o s t} = \int \frac{a c o s t d t}{a (s i n t + c o s t)}

= \int \frac{c o s t d t}{s i n t + c o s t}

= \int \frac{\frac{1}{2} \times 2 c o s t d t}{s i n t + c o s t}

= \int \frac{\frac{1}{2} \times (c o s t + c o s t) d t}{s i n t + c o s t}

= \int \frac{\frac{1}{2} \times (c o s t + s i n t + c o s t - s i n t) d t}{s i n t + c o s t}

= \int \frac{\frac{1}{2} (c o s t + s i n t) d t + \frac{1}{2} (c o s t - s i n t) d t}{c o s t + s i n t}

= \int \frac{1}{2} \times \frac{c o s t + s i n t}{c o s t + s i n t} d t + \int \frac{1}{2} \times \frac{c o s t - s i n t}{c o s t + s i n t} d t

= \int \frac{1}{2} d t + \int \frac{1}{2} \times \frac{c o s t - s i n t}{c o s t + s i n t} d t

l e t u = c o s t + s i n t, \frac{d u}{d t} = - s i n t + c o s t, d u = (c o s t - s i n t) d t

= \int \frac{1}{2} d t + \int \frac{1}{2} \times \frac{d u}{u}

= \frac{1}{2} t + l n (u) + C

= \frac{1}{2} t + l n (c o s t + s i n t) + C

B u t, x = a s i n t \dots . x = 0, a

0 = a s i n t \Rightarrow s i n t = 0 \Rightarrow t = {s i n}^{- 1} 0 \Rightarrow t = 0

A g a i n,

a = a s i n t \Rightarrow s i n t = 1 \Rightarrow t = {s i n}^{- 1} (1) \Rightarrow t = 90 \Rightarrow t = \frac{Π}{2}

{[\frac{1}{2} t + l n (s i n t + c o s t) + C]}_{0}^{\frac{Π}{2}}

= \frac{1}{2} (\frac{Π}{2}) + l n (s i n \frac{Π}{2} + c o s \frac{Π}{2}) + C - [\frac{1}{2} (0) + l n (s i n 0 + c o s 0) + C]

= \frac{Π}{4} + l n (1 + 0) + C - 0 - l n (0 + 1) - C

= \frac{Π}{4} + l n (1) + C - 0 - l n (1) - C

= \frac{Π}{4}

D O N E!

F o r c e \times D i s t a n c e = I S E

Commented by sandy_suhendra last updated on 01/Jul/16

{I t}^{'} s a g r e a t a n s w e r .

B u t I t h i n k t h e r e i s o n l y a l i t t l e m i s t a k e i n t y p i n g

s i n t = 1 \Rightarrow t = 90 (n o t 45) = \frac{π}{2}

Commented by Tawakalitu. last updated on 01/Jul/16

c o r r e c t e d . 10 Q