sinx-1-sinx-

Question Number 42579 by Raj Singh last updated on 28/Aug/18

\int s i n x / \sqrt{1 + s i n x}

Commented by maxmathsup by imad last updated on 28/Aug/18

l e t A = \int \frac{s i n x}{\sqrt{1 + s i n x}} d x w e h a v e 1 + s i n x = 1 + 2 s i n (\frac{x}{2}) c o s (\frac{x}{2})

= {(s i n (\frac{x}{2}) + c o s (\frac{x}{2}))}^{2} \Rightarrow \sqrt{1 + s i n x} = s i n (\frac{x}{2}) + c o s (\frac{x}{2})

A = \int \frac{1 + s i n x - 1}{\sqrt{1 + s i n x}} d x = \int \sqrt{1 + s i n x} d x - \int \frac{d x}{\sqrt{1 + s i n x}}

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

= - 2 c o s (\frac{x}{2}) + 2 s i n (\frac{x}{2}) - \int \frac{d x}{\sqrt{2} c o s (\frac{x}{2} - \frac{π}{4})} c h a n g e m e n t \frac{x}{2} - \frac{π}{4} = t g i v e

\int \frac{d x}{\sqrt{2} c o s (\frac{x}{2} - \frac{π}{4})} = \int \frac{2 d t}{\sqrt{2} c o s (t)} = \sqrt{2} \int \frac{d t}{c o s (t)}

=_{t a n (\frac{t}{2}) = u} \sqrt{2} \int \frac{1}{\frac{1 - u^{2}}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}} = 2 \sqrt{2} \int \frac{d u}{1 - u^{2}}

= \sqrt{2} \int (\frac{1}{1 + u} + \frac{1}{1 - u}) d u = \sqrt{2} l n ∣ \frac{1 + u}{1 - u} ∣ = \sqrt{2} l n ∣ \frac{1 + t a n (\frac{t}{2})}{1 - t a n (\frac{t}{2})} ∣

= \sqrt{2} l n ∣ t a n (\frac{t}{2} + \frac{π}{4}) ∣ = \sqrt{2} l n ∣ t a n (\frac{x}{4} - \frac{π}{8} + \frac{π}{4}) ∣= \sqrt{2} l n ∣ t a n (\frac{x}{4} + \frac{π}{8}) ∣ \Rightarrow

A = 2 s i n (\frac{x}{2}) - 2 c o s (\frac{x}{2}) - \sqrt{2} l n ∣ t a n (\frac{x}{4} + \frac{π}{8}) ∣ + c .

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Aug/18

\int \frac{s i n x}{\sqrt{1 + s i n x}} d x

\int \frac{1 + s i n x - 1}{\sqrt{1 + s i n x}} d x

\int \sqrt{1 + s i n x} d x - \int \frac{d x}{\sqrt{1 + s i n x}} d x

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

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

\int (s i n \frac{x}{2} + c o s \frac{x}{2}) d x - \frac{1}{\sqrt{2}} \int c o s e c (\frac{Π}{4} + \frac{x}{2}) d x

= - \frac{c o s \frac{x}{2}}{\frac{1}{2}} + \frac{s i n \frac{x}{2}}{\frac{1}{2}} - \frac{1}{\sqrt{2}} l n t a n (\frac{\frac{Π}{4} + \frac{x}{2}}{2}) + c

= 2 (- c o s \frac{x}{2} + s i n \frac{x}{2}) - \frac{1}{\sqrt{2}} l n t a n (\frac{Π}{8} + \frac{x}{4}) + c

Commented by maxmathsup by imad last updated on 28/Aug/18

y o u r a n s w e r i s c o r r e c t s i r T a n m a y .

Commented by tanmay.chaudhury50@gmail.com last updated on 28/Aug/18

t h a n k y o u s i r \dots