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If f(x) = {{:(0,",""where" x = (n)/(n+1...

If `f(x) = {{:(0,",""where" x = (n)/(n+1) n = 1"," 2"," 3"......"),(1,"," "else where"):}`,, then the value of `int_(0)^(2) f(x)dx`.

A

1

B

0

C

2

D

`oo`

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The correct Answer is:
C
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For each ositive integer n, define a function f _(n) on [0,1] as follows: f _(n((x)={{:(0, if , x =0),(sin ""(pi)/(2n), if , 0 lt x le 1/n),( sin ""(2pi)/(2n) , if , 1/n lt x le 2/n), (sin ""(3pi)/(2pi), if, 2/n lt x le 3/n), (sin "'(npi)/(2pi) , if, (n-1)/(n) lt x le 1):} Then the value of lim _(x to oo)int _(0)^(1) f_(n) (x) dx is:

RESONANCE-DEFINITE INTEGRATION & ITS APPLICATION -Exercise 2 Part - 1
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  2. lim(trarr(pi/2)^(-))int(0)^(t) tanthetasqrt(costheta)ln(costheta)d the...

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  3. If f(x) = {{:(0,",""where" x = (n)/(n+1) n = 1"," 2"," 3"......"),(1,...

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  4. If int0^ooe^(-x^2) dx=(sqrtpi)/2 , then int0^ooe^(-ax^2) dx where ...

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  5. If sum(i=1)^(4)(sin^(-1)x(i)+cos^(-1)y(i)) = 6 pi, then sum(i=1)^(4)x...

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  6. The tangent, represented by the graph of the function y=f(x), at the p...

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  7. Let A = int(0)^(1)(e^(t))/(1+t) dt, then int(a-1)^(a)(e^(-1))/(t-a-1) ...

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  8. int1^2 x^(2x^2+1)(1+2lnx)dx is equal to

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  9. If f(x) is a function satifying f(1/x) + x^(2)f(x) = 0 for all non-ze...

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  10. If C0/1+C1/2+C2/3=0 , where C0 C1, C2 are all real, the equation C2x...

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  11. If f(x) = int(0)^(x)(2cos^(2)3t+3sin^(2)3t)dt, f(x+pi) is equal to :

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  12. Let f(x) = int(0)^(x)(dt)/(sqrt(1+t^(2))) and g(x) be the inverse of ...

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  13. Let f(x) is differentiable function satisfying 2int(1)^(2)f(tx) dt ...

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  14. Let I(n) = int(0)^(1)x^(n)(tan^(1)x)dx, n in N, then

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  15. If u(n) = int(0)^(pi//2) x^(n)sinxdx, then the value of u(10) + 90 u(...

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  16. The value of int(1//e)^(tanx)(t)/(1+t^(2))dt+int(1//e)^(cotx)(1)/(t(1+...

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  17. Let A(1) = int(0)^(x)(int(0)^(u)f(t)dt) dt and A(2) = int(0)^(x)f(u).(...

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  18. lim(nrarroo) (sin'(pi)/(2n).sin'(2pi)/(2n).sin'(3pi)/(2n)"......."sin'...

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  19. Area bounded by the region consisting of points (x,y) satisfying y le ...

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  20. Find the area enclosed between the curves: y = loge (x + e) , x = loge...

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