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B
π8(β−α)
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C
π8(β−α)2
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D
None of these
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Solution
The correct option is Cπ8(β−α)2 Let I=∫βα√(x−α)(β−x)dx Put t=12(x−α+x−β)=x−12(α+β) So that (x−α)(β−x)=(t+c)(c−t)=c2−t2 Where c=12(β−α) Thus I=∫c−c√c2−t2dt=2∫c0√c2−t2dt =2[t2√c2−t2+c22sin−1tc]c0 =π8(β−α)2