Question

how can I integrate cosec x dx​

Answer 1

Answer:

log | Cosecx - Cotx | + C

Step-by-step explanation:

To find ---> Intregation of cosecx

Solution---> We have some formulee , which we used in calculating given intregation,

1) d/dx ( Cosecx ) = - Cosecx Cotx

2) d/dx ( Cotx ) = - Cosec²x

3) ∫ 1 / x dx = logx + C

Now returning to original problem,

∫ Cosecx dx

Multiplying and dividing by ( Cosecx - Cotx ) , we get,

= ∫ Cosecx ( Cosecx - Cotx ) dx / (Cosec x - Cotx )

= ∫(Cosec²x - Cosecx Cotx) dx / ( Cosecx - Cotx )

Let, ( Cosec x - Cotx ) = t

Differentiating with respect to x , both sides , we get,

=> d/dx ( Cosecx - Cotx ) = dt

=> { - Cosecx Cotx - (- Cosec²x ) } dx = dt

=> ( - Cosecx Cotx + Cosec²x ) dx = dt

=> ( Cosec²x - Cosecx Cotx ) dx = dt

Now

∫ dt / t

= log | t | + C

Putting value of t , we get,

= log |Cosecx - Cotx | + C

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Answer 2

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log |cosecx-cotx| +c

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