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Prove the following : int(0)^(1)e^(-x)co...

Prove the following : `int_(0)^(1)e^(-x)cos^(2)xdx lt int_(0)^(1)e^(-x^(2))cos^(2)xdx`

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Consider the integrals I_(1)=int_(0)^(1)e^(-x)cos^(2)xdx,I_(2)=int_(0)^(1) e^(-x^(2))cos^(2)x dx,I_(3)=int_(0)^(1) e^(-x^(2))dx and I_(4)=int_(0)^(1) e^(-(1//2)x^(2))dx . The greatest of these integrals, is

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Prove the following : e^(-(1)/(e)) lt int_(0)^(1)x^(x)dx lt 1

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RESONANCE ENGLISH-DEFINITE INTEGRATION & ITS APPLICATION -Self practive problem
  1. int(0)^(9) x(a^(2)-x^(2))^(7/2)dx

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  2. int(0)^(2) sqrt(2-x)dx.

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  3. Prove the following : int(0)^(1)e^(-x)cos^(2)xdx lt int(0)^(1)e^(-x^(2...

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  4. Prove the following : 0 lt int(0)^(pi//2)sin^(n+1)xdx lt int(0)^(pi//2...

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  5. Prove the following : e^(-(1)/(e)) lt int(0)^(1)x^(x)dx lt 1

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  6. Prove the following: -1/2lt=int0^1(x^3cosx)/(2+x^2)dx<1/2

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  7. Prove the following : 1 lt int(0)^(pi//2)sqrt(sinx)dx lt sqrt(pi/2)

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  8. Prove the following : 4/pi lt int(pi/4)^(pi/3) (tanx)/(x) lt (3sqrt(3)...

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  9. underset(nrarroo)lim[(1)/(n)+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+....

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  10. lim(nrarroo) [1/(1+n)+(1)/(2+n)+(1)/(3+n)+"....."+(1)/(5n)]

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  11. The value of lim(nrarroo) 1/n^2[sin^(3)'(pi)/(4n)+2sin^(3)'(2pi)/(4n)+...

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  12. lim(nrarroo) sum(r=0)^(n-1) (1)/(sqrt(n^(2)-r^(2)))

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  13. T h ev a l u eof(lim)(nvecoo)[t a npi/(2n)tan(2pi)/(2n)dottan(npi)/(2n...

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  14. Find the area bounded by the curves y = e^(x), y = |x-1| and x = 2.

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  15. Complete the area of the region bounded by the parabolas y^(2)+8x=16 ...

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  16. Find the area between the x-axis and the curve y = sqrt(1+cos4x), 0 ...

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  17. What is geometrical significance of (i) int(0)^(pi) |cosx| dx, (ii) ...

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  18. Find the area of the region bounded by the x-axis and the curves defin...

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  19. Find the area bounded by the curves x = y^(2) and x = 3-2y^(2).

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  20. Find the area bounded by the curve y = x^(2) - 2x + 5, the tangent to ...

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