Question

please answer this question​

Answer 1

$\bold{\huge{\underline{ Solution }}}$

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,

$\bold{\displaystyle\int}{\bold{\dfrac{ cos^{2}x}{1 + sinx}}}{\bold{dx}}$

By using trigonometric identity,

• $\sf{ sin^{2}{\theta}+ cos^{2}{\theta} = 1 }$

$\sf{\displaystyle\int}{\sf{\dfrac{ 1 -sin^{2}x }{1 + sinx}}}{\sf{dx}}$

By using algebraic identity :-

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

$\sf{\displaystyle\int}{\sf{\dfrac{ (1 + sinx) (1 - sinx) }{1 + sinx}}}{\sf{dx}}$

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

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

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.

Answer 2

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

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

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

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

Can be further written as

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

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

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

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 ;

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

This is the Required answer