Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
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
∫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
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
Register Yourself for a FREE Demo Class by Top IITians & Medical Experts Today !