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Show that: overset(1)underset(0)int (lo...

Show that: `overset(1)underset(0)int (log(1+x)/(1+x^2))dx= pi/8 log 2`.

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UNITED BOOK HOUSE-MISCELLANEOUS EXERCISE-EXERCISE
  1. Evaluate: overset(e)underset(1)int((x+1)^3)/(x^2) dx

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  2. Evaluate: overset(pi)underset(0)intxsinx cos^2xdx

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  3. Show that: overset(1)underset(0)int (log(1+x)/(1+x^2))dx= pi/8 log 2.

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  4. Evaluate: int(pi/4)^((3pi)/4)dx/(1+cosx)

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  5. Evaluate : underset(nrarrinfty)lim [1/n + 1/(n+1)+ 1/(n+2) + .... +1...

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  6. Solve: xdy-ydx= sqrt(x^2+y^2)dx.

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  7. Solvex dy + y dx = (ydx - xdy)/(x^2 + y^2)

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  8. Solvex cos x dy/dx + y (sin x + cos x) = 1

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  9. Solve(cos y + y cos x ) dx + sin (x - x sin y) dy = 0

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  10. Solve(d^2y)/(dt^2) = tan y sec ^2 y, given dy/dt = 0 when y = 0

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  11. Prove that cos (sinx) > sin (cos x) for all x in 0 le x le pi//2

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  12. If the line ax+by+c=0 is a normal to the curve xy=1 then

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  13. Prove that all points of the curve y^2=4a[x+asinx/a] at which the lang...

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  14. Find the point on the curve 4x^2 + a2y^2 = 4a^2, 4 < a^2 < 8 that is f...

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  15. The area bounded by the parabola y = x - x^2 and the line y = mx equal...

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  16. Prove that f(x)= sin x+sqrt 3 cosx has maximum value at x=pi/6

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  17. The length of the hypotenuse of a right angled triangle is 3 ft. Find ...

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  18. The angle between the tangents to the curves y=sin x and y = cos x at ...

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  19. Show that f(x) = (x-2)e^x + x + 2 is positive for all positive values ...

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  20. Vectors vecm, vecn, vecr are such that vecm + vecn + vecr = 0, prove t...

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