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Ifint0^(npi)f(cos^2x)dx=kint0^pif(cos^2x...

`Ifint_0^(npi)f(cos^2x)dx=kint_0^pif(cos^2x)dx ,t h e nfin dt h ev a l u ekdot`

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Verified by Experts

The correct Answer is:
`n`
Since `cos^(2)x` is a periodic function with period `pi` so is `f(cos^(2)x)`.
Hence `int_(0)^(pi) f(cos^(2)x)dx=n int_(0)^(pi) f(cos^(2)x)dx` or `k=n`
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CENGAGE PUBLICATION-DEFINITE INTEGRATION -CAE_TYPE
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  2. Evaluate: int0^(100pi)sqrt((1-cos2x))dxdot

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  3. Ifint0^(npi)f(cos^2x)dx=kint0^pif(cos^2x)dx ,t h e nfin dt h ev a l u ...

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  4. Evaluate int(0)^(npi+t)(|cosx|+|sinx|)dx, where n epsilonN and t epsil...

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  6. If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and pos...

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  7. Show that int0^(npi+v)|sinx|dx=2n+1-cosv , where n is a positive integ...

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  8. Ifint(pi/3)^xsqrt((3-sin^2t))dt+int0^ycostdt=0,then evaluate (dy)/(d...

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  9. Iff(x)=e^(g(x))a n dg(x)=int2^x(tdt)/(1+t^4), then find the value of ...

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  10. Evaluate (lim)(x to 4)int4^x((4t-f(t)))/((x-4))dt

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  11. Evaluate: lim(xto0)(int(0)^(x^(2))cos^(2)tdt)/(xsinx)

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  12. Find the points of minima for f(x)=int0^x t(t-1)(t-2)dt

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  13. Find the equation of tangent to y=int(x^2)^(x^3)(dt)/(sqrt(1+t^2))a t...

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  14. I f f(x)=int((pi^2)/(16))^(x^2)(sinxsinsqrt(theta))/(1+cos^2sqrt(theta...

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  15. If f(x) is a function satisfying f(x+a)+f(x)=0 for all x in R and pos...

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  16. Let f(x) be a differentiable function satisfying f(x)=int(0)^(x)e^((2t...

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  17. Ifint0^1(e^t dt)/(t+1)=a ,then find the value of int(b-1)^b(e^(-t)dt)/...

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  18. f(x)=int1^xlogt/(1+t+t^2)dt (xge1) then prove that f(x) =f(1/x)

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  19. f(x)=int1^x(tan^(-1)(t))/t dt \ AAx in R^+, then find the value of f...

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  20. Evaluate: int(sqrt(2)-1)^(sqrt(2)+1)((x^2-1))/((x^2+1)^2)dx

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