How do you integrate this? #int(1/(1+sinx+cosx))dx#

4 Answers
Feb 26, 2018

Break the fraction apart, solve the little pieces, then add them back together.

Explanation:

The fraction integrand can be separated into #int((1/1)+(1/sin(x))+(1/cos(x)))dx#. If an integrand can be separated, then all its parts can be solved separately.

This can be split into #int1dx# + #int(1/sin(x))dx# + #int(1/cos(x))dx#

Which is equivalent to

#intdx# + #intcsc(x)dx# + #intsec(x)dx#

#intdx# = #x#,
#intcsc(x)dx# = #ln|csc(x)-cot(x)| + C#,
#intsec(x)dx# = #ln|sec(x)+tan(x)| + C#

Now the three parts are added together.

#int(1/(1+sin(x)+cos(x)))dx# = #ln|csc(x)-cot(x)| + ln|sec(x)+tan(x)| +1 + C#

In order to calculate this integral you may use the following transform

#t=tan(x/2)#

hence

#sinx=(2t)/(1+t^2)# , #cosx=(1-t^2)/(1+t^2)# , #dx=2/(1+t^2)dt#

After some basic calculations which means just replace the above values to the integral and deduce you will get

#int 1/(1+sinx+cosx) dx=ln(abs(1+tan(x/2)))+c#

Feb 26, 2018

Kindly refer to the Explanation.

Explanation:

We have, #1+sinx+cosx=(1+cosx)+sinx#,

#=2cos^2(x/2)+2sin(x/2)cos(x/2)#,

#=2cos^2(x/2){1+sin(x/2)/cos(x/2)}#,

#=2/sec^2(x/2){1+tan(x/2)}#.

#:. I=int1/(1+sinx+cosx)dx#,

#=int(1/2*sec^2(x/2))/(1+tan(x/2))dx#,

#=int{d/dx(1+tan(x/2))}/(1+tan(x/2))dx#,

#=ln|1+tan(x/2)|+C," as Respected Konstantinos Michailidis Sir has already derived!"#

Enjoy Maths.!

Feb 26, 2018

# \quad \ int \ ( 1 /{ 1 + sin x + cos x } ) \ dx \ = \ ln \sqrt{ | { 1 + sin x } / { 1 + cos x } | } + C \quad. #

Explanation:

# "We want to find:" #

# \qquad \qquad \qquad int \ ( 1 /{ 1 + sin x + cos x } ) \ dx. #

# "We can proceed as follows:" #

# \qquad int \ 1 /{ 1 + sin x + cos x } \ dx #

# \qquad \qquad \quad \ = \ int \ ( 1 /{ 1 + sin x + cos x } ) cdot ( { 1 - ( sin x + cos x ) } /{ 1 - ( sin x + cos x ) } ) \ dx #

# \qquad \qquad \qquad = \ int \ { 1 - ( sin x + cos x ) } /{ ( 1 + sin x + cos x ) ( 1 - ( sin x + cos x ) ) ) \ dx #

# \qquad \qquad \qquad = \ int \ { 1 - ( sin x + cos x ) } /{ 1 - ( sin x + cos x )^2 ) \ dx #

# \qquad \qquad \qquad = \ int \ { 1 - ( sin x + cos x ) } /{ 1 - ( sin^2 x + 2 sinx cos x + cos^2 x ) } \ dx #

# \qquad \qquad \qquad = \ int \ { 1 - ( sin x + cos x ) } /{ 1 - ( sin^2 x + cos^2 x + 2 sin x cos x ) } \ dx #

# \qquad \qquad \qquad = \ int \ { 1 - ( sin x + cos x ) } /{ 1 - ( 1 + 2 sin x cos x ) } \ dx #

# \qquad \qquad \qquad = \ int \ { 1 - ( sin x + cos x ) } /{ color{red}cancel{1} - color{red}cancel{1} - 2 sin x cos x } \ dx #

# \qquad \qquad \qquad = \ - int \ { 1 - ( sin x + cos x ) } /{ 2 sin x cos x} \ dx #

# \quad = \ - int \ ( { 1 } /{ 2 sin x cos x } - { color{red}cancel{ sin x } } /{ 2 color{red}cancel{ sin x } cos x } - { color{red}cancel{ cosx } } /{ 2 sin x color{red}cancel{ cosx } } ) \ dx #

# \quad = \ - int \ ( { 1 } /{ sin 2 x } - { 1 } /{ 2 cos x } - { 1 } /{ 2 sin x } ) \ dx #

# \quad \ = \ - int \ ( csc 2 x - 1/2 sec x - 1/2 csc x ) \ dx #

# \quad \ = \ - ( - 1/2 ln | csc 2 x + cot 2 x | - 1/2 ln | sec x + tan x | #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + 1/2 ln | csc x + cot x | ) #

# \quad \ = \ 1/2 ( \ ln | csc 2 x + cot 2 x | + ln | sec x + tan x | #

# \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad - ln | csc x + cot x | \ ) #

# \quad \ = \ ln \sqrt{ { | csc 2 x + cot 2 x | cdot | sec x + tan x | } / { | csc x + cot x | } } #

# \quad \ = \ ln \sqrt{ { | { 1 + cos 2 x } / { sin 2 x } | cdot | { 1 + sin x } / { cos x } | } / { | { 1 + cos x } / { sin x } | } } #

# \quad \ = \ ln \sqrt{ | { { 1 + cos 2 x } / { sin 2 x } cdot { 1 + sin x } / { cos x } } / { { 1 + cos x } / { sin x } } | } #

# \quad \ = \ ln \sqrt{ | { 1 + cos 2 x } / { sin 2 x } cdot { 1 + sin x } / { cos x } cdot { sin x } / { 1 + cos x } | } #

# \quad \ = \ ln \sqrt{ | { ( sin^2 x + cos^2 x ) +( cos^2 x - sin^2 x ) } / { sin 2 x } cdot { 1 + sin x } / { cos x } cdot { sin x } / { 1 + cos x } | } #

# \quad \ = \ ln \sqrt{ | { color{red}cancel{ sin^2 x } + cos^2 x +cos^2 x - color{red}cancel{ sin^2 x } } / { sin 2 x } cdot { 1 + sin x } / { cos x } cdot { sin x } / { 1 + cos x } | #

# \quad \ = \ ln \sqrt{ | { color{red}cancel{ 2 } color{red}cancel{ cos^2 x } } / { color{red}cancel{ 2 } color{red}cancel{ sin x } color{red}cancel{ cos x } } cdot { 1 + sin x } / { color{red}cancel{ cos x } } cdot { color{red}cancel{ sin x } } / { 1 + cos x } | } #

# \quad \ = \ ln \sqrt{ | { 1 + sin x } / { 1 + cos x } | }\quad . #

# "So, at last !! :" #

# \quad \ int \ ( 1 /{ 1 + sin x + cos x } ) \ dx \ = \ ln \sqrt{ | { 1 + sin x } / { 1 + cos x } | } + C \quad. \quad \ square #