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I(1)=int(0)^((pi)/2)(sinx-cosx)/(1+sinxc...

`I_(1)=int_(0)^((pi)/2)(sinx-cosx)/(1+sinxcosx)dx, I_(2)=int_(0)^(2pi)cos^(6)dx`,
`I_(3)=int_(-(pi)/2)^((pi)/2)sin^(3)xdx, I_(4)=int_(0)^(1) In (1/x-1)dx`. Then

A

`I_(2)=I_(3)=I_(4)=0,I_(1)!=0`

B

`I_(1)=I_(2)=I_(3)=0,I_(4)!=0`

C

`I_(1)=I_(3)=I_(4)=0,I_(2)!=0`

D

`I_(1)=I_(2)=I_(3)=0,I_(4)!=0`

Text Solution

Verified by Experts

The correct Answer is:
C
`I_(1)=int_(0)^(pi//2)(sinx_cosx)/(1+sinx cos) dx`
`=int_(0)^(pi//2) ("sin"((pi)/2-x)-"cos"((pi)/2-x))/(1+"sin"((pi)/2-x)"cos"((pi)/2-x))`
`=int_(0)^(pi//2)(cosx-sinx)/(1+sinx cosx)dx=-I_(1)`
or `I_(1)=0`
`I_(3)=0` as `sin^(3)x` is odd
`I_(4)=int_(0)^(1)In((1-x)/x)dx`
`=int_(0)^(1)In((1-(1-x))/(1-x))dx`
`=int_(0)^(1)In x/(1-x)dx=-I_(4)`
or `I_(4)=0`
`I_(2)=int_(0)^(2pi)cos^(6)x dx=2int_(0)^(pi)cos^(6)x dx!=0`
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I_(1)=int_(0)^((pi)/(2))(sin x-cos x)/(1+sin x cos x)dx,I_(2)=int_(0)^(2 pi)cos^(6)xdx,I_(3)=int_((pi)/(2))^((pi)/(2))sin^(3)xdx,I_(4)=int_(0)^(1)1n((1)/(x)-1)dx. Then I_(1)=I_(3)=I_(4)=0,I_(1)!=0I_(1)=I_(3)=0,I_(4)!=0I_(1)=I_(2)=0,I_(4)!=0I_(1)=I_(2)=I_(3)=0,I_(4)!=0

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CENGAGE-DEFINITE INTEGRATION -Exercise (Single)
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  2. The value of int0^(2pi)[2 sin x] dx, where [.] represents the greate...

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  3. I(1)=int(0)^((pi)/2)(sinx-cosx)/(1+sinxcosx)dx, I(2)=int(0)^(2pi)cos^(...

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  4. Given int(0)^(pi//2)(dx)/(1+sinx+cosx)=A. Then the value of the defini...

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  5. "I f"I1=int(-100)^(101)(dx)/((5+2x-2x^2)(1+e^(2-4x))) "and"I2=int(-10...

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  6. The value of int0^oo(x dx)/((1+x)(1+x^2)) is equal to

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  7. For any integer n, the integral int0^pie^(cos^x)cos^3(2n+1)xdx has the...

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  8. Let f be a positive function. If I1 = int(1-k)^k x f[x(1-x)]\ dx and I...

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  9. Iff(x)=(e^x)/(1+e^x),I1=int(f(-a))^(f(a))xg(x(1-x)dx ,a n d I2=int(f(...

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  10. The value of int(1)^(2)(x^(2)+1)/(x^(4)-x^(2)+1)log(1+x-1/x)dx is

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  11. The value of the definite integral int0^(pi/2)sqrt(tanx)dx is sqrt(...

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  12. f(x)>0AAx in Ra n di sbou n d e ddotIf (lim)(nvecoo)[int0^a(f(x)dx)/...

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  13. Ifint0^1cot^(-1)(1-x+x^2)dx=lambdaint0^1tan^(-1)x dx ,t h e nlambdai s...

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  14. The value of the definite integral int(-1)^(1)(1+x)^(1//2)(1-x)^(3//2)...

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  15. The value of the integral int(-3pi//4)^(5pi//4)((sinx+cosx))/(e^(x-pi/...

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  16. I(1)=int(0)^((pi)/2)In (sinx)dx, I(2)=int(-pi//4)^(pi//4)In(sinx+cosx)...

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  17. IfI1=int0^(pi/2)(cos^2x)/(1+cos^2x)dx ,I2=int0^(pi/2)(sin^2x)/(1+sin^2...

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  18. int0^(pi/2) (xsinxcosx)/(cos^4x+sin^4x) dx=

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  19. For x epsilonR, and a continuous function f let I(1)=int(sin^(2)t)^(1+...

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  20. Ifint(-pi)^((3pi)/4)(e^(pi/4)dx)/((e^x+e^(pi/4))(sinx+cosx)=kint(-pi/2...

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