Question

integral(1-cosx) cosec^2x​

Answer 1

Answer:

sin²x/sin²x/2. cos²x/2 + c

Step-by-step explanation:

i (1 - cosx)cosec²x

i (cosec²x - cosx. cosec²x)

i cosec²x.dx - i cosx/sin²x

- cotx - i dt/t² [let sinx = t, and cosx.dx= dt]

- cotx - 1/t + c

- cotx - 1/sinx + c

- cotx + cosecx + c

-cosx/sinx + 1/sinx + c

(1 - cosx)/sinx + c

(1 - 1 + 2sin²x)/sinx + c

2sin²x/sinx + c

2sin²x/(2. sin²x/2. cos²x/2) + c

sin²x/sin²x/2. cos²x/2 + c

Answer 2

Answer

• csc x - cot x + c

Given

$\rm \bullet \;\; \int (1-cosx)csc^22x\ dx$

To Find

$\rm Integral\ value$

Solution

$\rm \int (1-cosx)csc^22x\ dx\\\\\implies \rm \int csc^22x\ dx -\int csc^22x.cosx\ dx\\\\\implies \rm -cotx\ -\int \dfrac{cosx}{sin^2x}dx...(1)$

Let , u = sin x

⇒ du = cos x dx

Sub. this in (1) , we get ,

$\implies \rm -cotx-\int \dfrac{du}{u^2}\\\\\implies \rm -cotx-\bigg( -\dfrac{1}{u}\bigg)+c\\\\\implies \rm -cotx+\dfrac{1}{u}+c\\\\\implies \rm -cotx+\dfrac{1}{sinx}+c\\\\\implies \rm -cotx+cscx+c\\\\\implies \rm cscx-cotx+c$

More Info

• d/dx ( sin x ) = cos x
• d/dx ( cos x ) = - sin x
• d/dx ( tan x ) = sec² x
• d/dx ( cot x ) = - csc² x
• d/dx ( sec x ) = sec x . tan x
• d/dx ( csc x ) = - csc x . cot x