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Equate The Integrals, Get Constant (Posted on 20071024) 

Determine the value of a constant C such that:
∫_{0}^{pi/3}(sin y/cos^{2}y)dy = ∫_{0}^{C}(√(z+C)  √z)^{1}dz
Note: The range of the first integral reads 'pi/3'.

Submitted by K Sengupta

Rating: 2.5000 (2 votes)


Solution:

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C = 9/32
EXPLANATION:
Substituting t = sec y, we obtain dt = sec y .tan y.dy
Thus, the integral on the right is:
integral(dt), t = 1 to 2
= 21 = 1
Also, the integral on the left:
= integral (V(z+C) – VC)^{1}.dz, z = 0 to C
= (1/C)* integral (V(z+C) + VC)^{1}.dz, z = 0 to C
= 4*(V2/3)*VC
Thus, 32*C/9 = 1, giving C = 9/32, so that:
The required value of C is 9/32.

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