Home
Class 12
MATHS
If int(pi/4)^((3pi)/4) x/(1+sinx)dx=k(sq...

If `int_(pi/4)^((3pi)/4) x/(1+sinx)dx=k(sqrt(2)-1)`, then `k`= (A) `0` (B) `pi` (C) `2pi` (D) none of these

Text Solution

Verified by Experts

The correct Answer is:
N/a
Let `I = int_(0)^(pi) (x)/(1+sinx)dx`
and `I = int_(0)^(pi) (pi-x)/(1+sin(pi-x)) dx = int_(0)^(x) (pi-x)/(1+sinx) dx`
In adding Eqs. (i) and (ii), we get
`2I = pi int_(0)^(pi) (1)/(1+sinx) dx`
`= piint_(0)^(pi) = ((1-sinx)dx)/((1+sinx)(1+sinx))`
`= piint_(0)^(pi) ((1-sinx)dx)/(cos^(2)x)`
`= pi int_(0)^(pi) (sec^(2)x-tanx.secx)dx`
`= piint_(0)^(pi)sec^(2)xdx - pi int_(0)^(pi)sec x x. tan x dx`
`= pi [tanx]_(0)^(pi) - pi [secx]_(0)^(pi)`
`= pi [tanx - sex x ]_(0)^(x)`
`= pi [tanpi - secpi - tan 0 - sec 0]`
`rArr 2I = pi[0+1-0+1]`
`2I = 2pi`
`:. I = pi`
Promotional Banner

Topper's Solved these Questions

  • DIFFERENTIAL EQUATIONS

    NCERT EXEMPLAR|Exercise Differential Equations|77 Videos

  • INVERSE TRIGONOMETRIC FUNCTIONS

    NCERT EXEMPLAR|Exercise Inverse Trigonometric Functions|55 Videos

Similar Questions

Explore conceptually related problems

int_(pi//4)^(3pi//4)(xsinx)/(1+sinx)dx

int_(-pi//4)^(pi//4)(dx)/((1+sinx))

int_(-pi//4)^(pi//4)|sinx|dx=(2-sqrt(2))

int_(0)^(pi)(x sinx)/((1+sinx))dx=pi((pi)/(2)-1)

sqrt(3)int_0^pi dx/(1+2sin^2x)= (A) -pi (B) 0 (C) pi (D) none of these

If int_(0)^(20)sqrt(1-cos pi x)dx=(10k sqrt(2))/(pi), then k is

int_-pi^(3pi) cot^-1(cotx)dx= (A) pi^2 (B) 2pi^2 (C) 3pi^2 (D) none of these

The value of the integral int_(-(3 pi)/(4))^((pi)/(4))((sin x+cos x)/(e^(x-(pi)/(4))+1))dx(A)0(B)1(C)2(D) none of these

If I=int_(-pi)^pi e^(sinx)/(e^(sinx)+e^(-sinx))dx , then I equals (A) pi/2 (B) 2pi (C) pi (D) pi/4

If int_0^pi x f(sinx) dx=A int_0^(pi/2) f(sinx)dx , then A is (A) pi/2 (B) pi (C) 0 (D) 2pi