Question

Please answer this question​

Answer 1

We are given with an integral and need to solve the integral , so let's start ;

`{:\implies \quad \displaystyle \int \sf (\cos^(2)(x))/(1+\sin (x))dx}`

As we know that sin²(x) + cos²(x) = 1 , using this

`{:\implies \quad \displaystyle \int \sf (1-\sin^(2)(x))/(1+\sin (x))dx}`

Can be further written as

`{:\implies \quad \displaystyle \int \sf (1^(2)-\sin^(2)(x))/(1+\sin (x))dx}`

`{:\implies \quad \displaystyle \int \sf \frac{\cancel{\{1+\sin (x)\}}\{1-\sin (x)\}}{\cancel{\{1+\sin (x)\}}}dx\quad \qquad \{\because a^(2)-b^(2)=(a+b)(a-b)\}}`

`{:\implies \quad \displaystyle \int \sf \{1-\sin (x)\}dx}`

Now , as integrals follow distributive property , so ;

`{:\implies \quad \displaystyle \int \sf 1\: dx-\int \sin (x)dx}`

Now , as antiderivative (Integration) of sin(x) is -cos(x) + C and that of dx is x + C So ;

`{:\implies \quad \displaystyle \bf \therefore \underline{\underline{\int \bf (\cos^(2)(x))/(1+\sin (x))=x+\cos (x)+C}}}`

This is the Required answer

Answer 2

`\bold{\huge{\underline{ Solution }}}`

Here, we will use the concept of integration and algebric identities

• Integration is the process of finding function that is a derivate of given function
• Three important trigonometric identities :-

1. 
   `\sf{ sin^(2){\theta}+ cos^(2){\theta} = 1 }`
2. 
   `\sf{ 1 + tan^(2){\theta} = sec^(2){\theta} }`
3. 
   `\sf{ 1 + cot^(2){\theta} = cosec^(2){\theta} }`

Let's Begin :-

We have,

`\bold{\displaystyle\int}{\bold{( cos^(2)x)/(1 + sinx)}}{\bold{dx}}`

By using trigonometric identity,

• 
  `\sf{ sin^(2){\theta}+ cos^(2){\theta} = 1 }`

`\sf{\displaystyle\int}{\sf{( 1 -sin^(2)x )/(1 + sinx)}}{\sf{dx}}`

By using algebraic identity :-

• 
  `\sf{ a^(2) - b^(2) = ( a + b) (a - b) }`

`\sf{\displaystyle\int}{\sf{( (1 + sinx) (1 - sinx) )/(1 + sinx)}}{\sf{dx}}`

`\sf{\displaystyle\int}{\sf{( 1 - Sinx)dx}}`

`\sf{ {\displaystyle\int} dx - {\displaystyle\int}sinxdx}`

We know that,

• 
  `\sf{{\int\displaystyle}sin{\theta} d{\theta} = - cos {\theta} + C }`

`\sf{ x - (-cosx) + c}`

`\bold{ x + cosx + c}`

Hence, The answer is x + Cosx + c.