calculate-0-pi-2-ln-cosx-sinx-ex-

Question Number 47112 by maxmathsup by imad last updated on 04/Nov/18

c a l c u l a t e \int_{0}^{\frac{π}{2}} l n (c o s x + s i n x) e x

Commented by tanmay.chaudhury50@gmail.com last updated on 05/Nov/18

Commented by tanmay.chaudhury50@gmail.com last updated on 05/Nov/18

0 < \int_{0}^{\frac{π}{2}} l n (s i n x + c o s x) < \frac{π}{2} \times 0.5

0 < I < \frac{π}{4}

Commented by prof Abdo imad last updated on 06/Nov/18

l e t I = \int_{0}^{\frac{π}{2}} l n (c o s x + s i n x) d x \Rightarrow

I = \int_{0}^{\frac{π}{2}} l n (\sqrt{2} c o s (x - \frac{π}{4})) d x = \frac{π}{4} l n (2) + \int_{0}^{\frac{π}{2}} l n (c o s (x - \frac{π}{4})) d x

b u t \int_{0}^{\frac{π}{2}} l n (c o s (x - \frac{π}{4})) d x =_{\frac{π}{4} - x = t} - \int_{\frac{π}{4}}^{- \frac{π}{4}} l n (c o s t) d t

= \int_{- \frac{π}{4}}^{\frac{π}{4}} l n (c o s t) d t = 2 \int_{0}^{\frac{π}{4}} l n (c o s t) d t

l e t A = \int_{0}^{\frac{π}{4}} l n (c o s t) d t a n d B = \int_{0}^{\frac{π}{4}} l n (s i n t) d t

A + B = \int_{0}^{\frac{π}{4}} l n (c o s t s i n t) d t = \int_{0}^{\frac{π}{4}} l n (\frac{s i n (2 t)}{2}) d t

= - \frac{π}{4} l n (2) + \int_{0}^{\frac{π}{4}} l n (s i n (2 t)) d t

=_{2 t = u} - \frac{π}{4} l n (2) + \frac{1}{2} \int_{0}^{\frac{π}{2}} l n (s i n u) d u

= - \frac{π}{4} l n (2) - \frac{π}{4} l n (2) = - \frac{π}{2} l n (2) a l s o

A - B = \int_{0}^{\frac{π}{4}} l n (c o s t) d t - \int_{0}^{\frac{π}{4}} l n (s i n t) d t b u t

\int_{0}^{\frac{π}{4}} l n (s i n t) d t =_{t = \frac{π}{2} - x} - \int_{\frac{π}{2}}^{\frac{π}{4}} l n (c o s x) d x

= \int_{\frac{π}{4}}^{\frac{π}{2}} l n (c o s x) d x = \int_{0}^{\frac{π}{2}} l n (c o s x) d x - \int_{0}^{\frac{π}{4}} l n (c o s x) d x

= - \frac{π}{2} l n (2) - A \Rightarrow A - B = - \frac{π}{2} l n 2 - A \Rightarrow

2 A = B - \frac{π}{2} l n (2) \Rightarrow 2 A = - \frac{π}{2} l n (2) - A - \frac{π}{2} l n (2)

\Rightarrow 2 A = - A - π l n (2) \Rightarrow 3 A = - π l n (2) \Rightarrow

A = - \frac{π}{3} l n (2) \Rightarrow I = - \frac{2 π}{3} l n (2) .

Answered by tanmay.chaudhury50@gmail.com last updated on 05/Nov/18

l n (c o s x + s i n x)

\int_{0}^{\frac{π}{2}} l n {\sqrt{2} \times (\frac{1}{\sqrt{2}} s i n x + \frac{1}{\sqrt{2}} c o s x) d x

\frac{1}{2} l n 2 \int_{0}^{\frac{π}{2}} d x + \int_{0}^{\frac{π}{2}} l n {s i n (\frac{π}{4} + x)} d x

I_{1} = (\frac{1}{2} l n 2) \times \frac{π}{2} = \frac{π}{4} l n 2

I_{2}

w a i t \dots