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If In=int0^(pi//4)tan^n x dx , show t...

If `I_n=int_0^(pi//4)tan^n x dx` , show that `1/(I_2+I_4),1/(I_3+I_5),1/(I_4+I_6),1/(I_5+I_7),` form an A.P. Find the common difference of this progression.

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BANSAL-INTEGRALS-All Questions
  1. If int0^(10)f(x)dx=5, find the value of sum(k=1)^(10)int0^1f(k-1+x)dxd...

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  2. If y(x)=int(pi^2/16)^(x^2) (cosx cos sqrt(theta))/(1+sin^2 sqrt(theta)...

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  3. If In=int0^(pi//4)tan^n x dx , show that 1/(I2+I4),1/(I3+I5),1/(I4+...

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  4. Evaluate: lim(n->oo)[1/((n+1)^2\ )+1/((n+2)^2)+...+1/((2n)^2)]

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  5. Evaluate: inta^b e^x dx using limit of sum

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  6. Evaluate : (lim)(n->oo)sum(r=1)^n1/((n^2+r^2)^(1//2))

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  7. Evaluate the following definite integrals: int0^(pi//4)(sqrt(1-sin2...

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  8. Evaluate the following definite integrals: int0^(pi//2)xsinx sin2x d...

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  9. int0^(2pi)(dx)/(2+sin2x)

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  10. Evaluate the following definite integrals: int(1//e)^e|l\ n x|dx

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  11. Evaluate the following definite integral: int0^(pi//2)(sinx+cos x)dx

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  12. Evaluate the following definite integrals: int0^1(e^(-x))/(1+e^x)dx

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  13. Evaluate the following definite integrals: int0^(pi//4)(sinx+cosx)/...

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  14. The value of int-pi^pi cos^2x/[1+a^x].dx,a>0 is

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  15. For a positive integer n, let I (n) int (-pi)^(pi) ((pi)/(2) -|x|) cos...

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  16. Show that : int0^1(logx)/((1+x))dx=-int0^1(log(1+x))/x dx

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  17. Find the limit, when n->oo,\ of\ [((n^3+1^3)(n^3+2^3)(n^3+3^3).........

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  18. Find the function f(x) if it satisfies the condition f(x)=x+int0^(...

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  19. If int0^1 sinx/(1+x)dx=K the the value of int(4pi-2)^(4pi)sin(x/2)/...

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  20. I=int(0)^(2)(3x^(2)-3x+1)cos(x^(3)-3x^(2)+4x-2)dx

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