MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Menu
Manish Prasad Grade:
        


Hello there:


I have been trying to do the following integral but have not been successful. Could someone please lend a hand?


1 / (sin(x) + sec(x))


Thanks a ton!
Manish

7 years ago

Answers : (2)

askiitianexpert arulmani
6 Points
										

∫dx/(sin x+sec x)=∫dx/(sin x + (1/cos x))=∫cos x dx /(1+sin x cos x)


Multiplying by 2, we get = ∫2 cos x dx / (2 + 2 sin x cos x) 


2 cos x dx can be substituted with (cos x + sin x) + (cos x - sin x) and


2 + 2 sin x cos x can be substituted with either (3 - (sin x - cos x)2) or (1 + (sin x + cos x)2)


Hence we get,


∫2 cos x dx / (2 + 2 sin x cos x) = ∫ (cos x + sin x) dx / (3 - (sin x - cos x)2) + ∫(cos x - sin x) dx / (1 + (sin x + cos x)2)


FIRST PART OF THE INTEGRAL


∫ (cos x + sin x) dx / (3 - (sin x - cos x)2)


Integrate by substitution, put y = sin x - cos x


then dy = (cos x +sin x) dx,


and we get ∫ dy / (3 - y2) = (1/(2*√3)) [ ∫ dy / (√3 + y) +  ∫ dy / (√3 - y) ]


=> (1/(2*√3)) [ log(√3 + y) -  log(√3 - y) ] + C


replacing y with its original value, we get


=> (1/(2*√3)) [ log(√3 + (sin x - cos x)) -  log(√3 - (sin x - cos x)) ] + C



SECOND PART OF THE INTEGRAL


∫(cos x - sin x) dx / (1 + (sin x + cos x)2)


Integrate by substitution, put z = sin x + cos x


then dz = (cos x - sin x) dx,


and we get ∫dz / (1 + (z)2)


Let z = tan Θ, then dz = sec2Θ dΘ


∫dz / (1 + (z)2) = ∫sec2Θ dΘ / (1 + (tan Θ)2) = ∫sec2Θ dΘ / sec2Θ = ∫dΘ = Θ = tan-1z + C


replacing z with its original value, we get, 


=> tan-1(sin x + cos x)+ C



FINAL ANSWER


Now adding the first and second parts, we get the result as


=> (1/(2*√3)) [ log(√3 + (sin x - cos x)) -  log(√3 - (sin x - cos x)) ] + tan-1(sin x + cos x)+ C



7 years ago
Anurag Kishore
37 Points
										

Hi, the integral is









 

∫ cosx dx / (sinx + cosx)


= 1/2 [2 cosx dx /(sinx + cosx)


= 1/2 [(cosx + sinx) + (cosx - sinx) dx] / (sinx + cosx)


Now divide individually


put sinx + cosx = t and integrate


 


Final answer


 


x/2  +  1/2  log I sinx + cosx I  + c


6 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details