Question

please help with this integration

Answer 1

then later integrate it by parts

Answer 2

Answer:

$\to \sf \displaystyle \int \sf \dfrac{1}{cosx-sinx} dx$

Multiply and divide the numerator by √2.

$\to \sf \displaystyle \int \sf \dfrac{1}{\dfrac{\sqrt{2} }{\sqrt{2} }(cosx-sinx)} dx$

$\to \sf \dfrac{1}{\sqrt{2} } \displaystyle \int \sf \dfrac{1}{\dfrac{1}{\sqrt{2}}cosx-\dfrac{1}{\sqrt{2}}sinx} dx$

⇒ cos(45) = 1/√2, sin(45)=1/√2

$\to \sf \dfrac{1}{\sqrt{2} } \displaystyle \int \sf \dfrac{1}{sin(45^{\circ})cosx-cos(45^{\circ})sinx} dx$

⇒ sin(a-b)=sina cosb-sinb cosa

Here, a = 45°, b = x. Using the above Identity,

$\to \sf \dfrac{1}{\sqrt{2} } \displaystyle \int \sf \dfrac{1}{sin(45-x)} dx$

$\to \sf \dfrac{1}{\sqrt{2} } \displaystyle \int \sf cosec(45-x) dx$

⇒ Substitute 45 - x = t. dx = -dt.

$\to \sf \dfrac{-1}{\sqrt{2} } \displaystyle \int \sf cosec(t) dt$

We know that ∫cosec x =

$\to \sf \dfrac{-1}{\sqrt{2} } ( ln|cosec \: t - cot \: t|)$

Substitute t back.

Answer:

$\to \boxed{\pink{\bf{\dfrac{-1}{\sqrt{2} } ( ln|cosec \: (45-x) - cot \: (45-x)|)}}}$