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Prove the following : e^(-(1)/(e)) lt in...

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

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RESONANCE-DEFINITE INTEGRATION & ITS APPLICATION -Self practive problem
  1. Prove the following : int(0)^(1)e^(-x)cos^(2)xdx lt int(0)^(1)e^(-x^(2...

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

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

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

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

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

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  7. lim(nrarroo) {1/n+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+"......"+1/(...

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

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  9. lim(nrarroo) [sin^(3)'(pi)/(4n)+2sin^(3)'(2pi)/(4n)+3sin^(3)'(3pi)/(4n...

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

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  11. lim(nrarroo) (tan'(pi)/(2n)tan'(2pi)/(2n)tan'(3n)/(2n)"......."tan'(np...

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

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

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

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

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  16. Find the area of the region bounded by the x-axis and the curves defi...

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

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

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  19. Find the area of the region bounded by the curves y=x-1 & (y-1)^2=4(x+...

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  20. Find the area of the region lying in the first quadrant and included b...

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