Flag Integral Calculus> Integrate...
question mark

Evaluate the following integrals:-

Q-1)∫1/√1+cosx dx

Q-2)∫{1+TANXTAN(X+θ)} dx

Q-3)∫sin(x-α)/sin(x+α) dx

Q-4)∫secxtanx/3x+5 dx

Nimish Singh , 11 Years ago
Grade 11
anser 4 Answers
Akash Kumar Dutta

Last Activity: 11 Years ago

in the third put x+a = t ...x=t-a
dx=dt
the eq becomes.
  int. [sin (t-2a)/sint] dt
  int. [cos 2a - cot t.sin2a ] dt
  =cos 2a.t - sin 2a.log |sin t| + C
  = (x+a)cos 2a - sin 2a.log |sin (x+a) | +C...ANS

Akash Kumar Dutta

Last Activity: 11 Years ago

for the first,
we know 1+cosx=2cos ^2 x/2
hence the eq. becomes,
int [ 1/(root2)cos (x/2)]dx
= 1/root2 . int [ 1/cos (x/2)]dx...now put x/2=m....such that dx=2dm
= 1/root2 .int [2secm]dm
=(root 2).log |sec m + tan m | + C
= (root2).log | sec x/2 + tan x/2 | +C...ANS

Akash Kumar Dutta

Last Activity: 11 Years ago

for the SECOND...
i had difficulty using theta here so in the midway i have taken (o) as theta....so dont get confused!!!
we know,
tan
θ=tan(x+θ-x)=tan(x+θ)-tanx/ 1+tan x.tan(n+θ)

hence 1 + tanxtan(x+
θ)=1/tan (o) [ tan(x+(o) ) - tanx]
hence the eq,. becomes,
1/tan(o) int [ tan( x+(o) ) - tanx ]
1/tan (o) [ -log |cos ( x+(o) ) + log|cosx| ]
1/tan(o) [ log |cos x / cos ( x+(o) ) ]

= log | cos x / cos ( x + (o) ) |
   ______________________     + C       ........ANS
                  tan(o)

Akash Kumar Dutta

Last Activity: 11 Years ago

Sorry man not to able to solve thos one....but i think we need to proceed like this
these types of sums are impossible without int by parts.....
we know the integral of secxtanx=secx so take it as v,
and 1/3x+5 = u
then by parts try to integrate
hope it helps...

Provide a better Answer & Earn Cool Goodies

Enter text here...
star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Ask a Doubt

Get your questions answered by the expert for free

Enter text here...