Home
Class 12
MATHS
intsqrt(1+cos e cx)dx equals (a)2sin^...

`intsqrt(1+cos e cx)dx` equals
(a)`2sin^(-1)sqrt(sinx)+c` (b) `sqrt(2)cos^(-1)sqrt(cosx)+c` (c) `c-2sin^(-1)(1-2sinx)` (d) `cos^(-1)(1-2sinx)+c`

A

`2sin^(-1)sqrt(sinx)+c`

B

`sqrt(2)cos^(-1)sqrt(cosx)+c`

C

`c-2sin^(-1)(1-2sinx)`

D

`cos^(-1)(1-2sinx)+c`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`I=int(sqrt((1+sinx)(1-sinx)))/(sqrt(sinx(1-sinx)))dx`
`=int(cosx)/(sqrt(sinx(1-sinx)))dx`
`=int(cosx)/(sqrt((1)/(4)-((1)/(2)-sinx)^(2)))dx `
`=int(-dt)/(sqrt(((1)/(2))^(2)-t^(2))) " " ("Putting " (1)/(2)-sinx=t)`
`=-sin^(-1)((t)/(1//2))+C`
`=-sin^(-1)(1-2sinx)+C`
`=cos^(-1)(1-2sinx)+C-(pi)/(2) `
`=cos^(-1)(1-2sinx)+C`
`=cos^(-1)(1-2(sqrt(sinx))^(2))+C`
`=cos^(-1)(1-2sin^(2)t)+C " " ("Putting " sqrt(sinx)=sint)`
`=cos^(-1)(cos2t)+C`
`=2t+C " " [:' sqrt(sinx)gt0 implies sint gt0 impliest in(0(pi)/(2))]`
`=2sin^(-1)(sqrt(sinx))+C`
Promotional Banner

Topper's Solved these Questions

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Comprehension)|17 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Matrix)|4 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Exercise (Single)|77 Videos

  • HYPERBOLA

    CENGAGE|Exercise JEE Advanced Previous Year|14 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

Similar Questions

Explore conceptually related problems

y = sin^(-1)(x/sqrt(1+x^2)) + cos^(-1)(1/sqrt(1+x^2))

If f(x)={sin^(-1)(sinx),xgt0 (pi)/(2),x=0,then cos^(-1)(cosx),xlt0

If tanx=b/a , then sqrt((a+b)/(a-b))+sqrt((a-b)/(a+b)) is equal to (a) 2sinx//sqrt(sin2x) (b) 2cosx//sqrt(cos2x) (c) 2cosx//sqrt(sin2x) (d) 2sinx//sqrt(cos2x)

If -1< x < 0 , then cos^(-1)x is equal to (a) sec^(-1)(1/ x) (b) pi-sin^(-1)sqrt(1+x^2) (c) pi+tan^(-1)(x/(sqrt(1-x^2))) (d) cot^(-1)(x/(sqrt(1-x^2))) .

d/(dx)(cos^(-1)sqrt(cosx)) is equal to (a) 1/2sqrt(1+s e cx) (b) sqrt(1+secx) (c) -1/2sqrt(1+secx) (d) -sqrt(1+secx)

The value of the integral int(cos^3x+cos^5x)/(sin^2x+sin^4x)dxi s (a). sinx-6tan^(-1)(sinx)+C (b). sinx-2(sinx)^(-1)+C (c). sinx-2(sinx)^(-1)-6tan^(-1)(sinx)+C (d). sinx-2(sinx)^(-1)+5tan^(-1)(sinx)+C

Find the sum cos e c^(-1)sqrt(10)+cos e c^(-1)sqrt(50)+cos e c^(-1)sqrt(170)++cos e c^(-1)sqrt((n^2+1)(n^2+2n+2))

int(dx)/(e^xsqrt(2e^x-1))= 2sec^(-1)sqrt(2e^x)+c -2tan^(-1)1/(sqrt(2e-1))+c 2sec^(-1)(sqrt(2)e^x)+c (d) (2sqrt(2e^x-1))/2e^x 2tan^(-1)sqrt(2e^x-1)+c

intsqrt(e^x-1)dxi se q u a lto 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)+c sqrt(e^x-1)+tan^(-1)sqrt(e^x-1)+c 2[sqrt(e^x-1)-tan^(-1)sqrt(e^x-1)]+c

If sin^(-1)x_i in [0,1]AAi=1,2,3, .28 then find the maximum value of sqrt(sin^(-1)x_1)sqrt(cos^(-1)x_2)+sqrt(sin^(-1)x_2)sqrt(cos^(-1)x_3)+ sqrt(sin^(-1)x_3)sqrt(cos^(-1)x_4)++sqrt(sin^(-1)x_(28))sqrt(cos^(-1)x_1)