Question

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

Answer 1

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`

Answer 2

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`

Answer 3

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.!

Answer 4

`\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`