Question

Integrate the Question in attachment!​

Answer 1

To integrate :

$\star \: \int \: \dfrac{ \sin \: x - \cos \: x}{ \sin \: x \: + \cos \: x} dx$

Here ,

• We use substitution method ;

Let

• Sin x + cos x = t

Now differentiate w.r.t x both sides :

$\implies \: \frac{d}{dx} ( \sin \: x + \cos \: x) = \frac{dt}{dx} \\ \\ \implies (\cos \: x \: - \sin \: x)dx = dt \\ \\ \implies \: - ( \sin \: x \: - \cos \: x)dx = dt \\ \\ \implies( \sin \: x - \cos \: x)dx = - dt$

put the values :

$\implies \: - \int \: \dfrac{dt}{t} \\ \\ \implies \: - log(t) + c$

Now , put the value of t .

$\implies \: - log( \sin \: x + \cos \: x) + c.$

Formula used :

$\star \bold{ \int \: \dfrac{1}{x} dx = log(x) + c} \\ \\ \star \bold{ \dfrac{d}{dx} \sin \: x = \cos \: x}$