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Prove that overset(pi//2)underset(0)intl...

Prove that `overset(pi//2)underset(0)intlog(sinx)dx = -pi/2 log2`

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UNITED BOOK HOUSE-SET 1-EXERCISE
  1. Prove that the equation of the plane which passes through the point (2...

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  2. Show that the probability that exactly one of the events A and B occur...

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  3. Show that tan^(-1)(1/sqrt3tanx/2)=1/2 cos^(-1)(1+2cosx)/(2+cosx).

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  4. If A = [[1,x,-2],[2,2,4],[0,0,2]] and A^2+2I3=3A Find x, here I3 is th...

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  5. Prove that . |[1,a,a^2-bc],[1,b,b^2-ca],[1,c,c^2-ab]|=0

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  6. Show that , |[(a^2+b^2)/c,c,c],[a,(b^2+c^2)/a,a],[b,b,(c^2+a^2)/b]|=4a...

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  7. If f(x)= tan^(-1)(x/(1+20x^2)) show that f'(x)=5/(1+25x^2)-4/(1+16x^2)...

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  8. Let y = (sin^(-1)x)^2+(cos^(-1)x)^2 show that (1-x^2)(d^2y)/(dx^2)-x(d...

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  9. Evaluate: int (dx)/sqrt(2/3x^3-x^2+1/3)

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  10. Solve: (6x+9y-7)dx= (2x+3y-6)dy

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  11. Solve: (1-x^2)(dy)/(dx)-xy=x^2, given y = 2 when x= 0.

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  12. If vec(AB) = 2 hati- 4 hatj +5 hatk and vec(BC)= hat i-2 hatj-3hatk in...

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  13. Find x such that the four points A(3,2,1),B(4,x,5),C(4,2,-2) and D(6,5...

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  14. Prove that overset(pi//2)underset(0)intlog(sinx)dx = -pi/2 log2

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  15. Urn A contains 1 white, 2 black and 3 red balls urn B contains 2 white...

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  16. A cylindrical tin can, open at the top, of a given capacity has to be...

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  17. If the straight line y =kx+3 is a tangent to the hyperbola 7x^2-4y^2=2...

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  18. Using calculus, find the area bounded by the curve |x|+|y|=1.

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  19. Show that the lines vecr= (hat i+hat j+hat k)+t(hat i-hat j+hat k) and...

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  20. Find the equation of the plane passing through the points (-1,1,1) an...

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