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`int_(5/2)^5(sqrt((25-x^2)^3))/(x^4)dxi se q u a lto``pi/6`(b) `(2pi)/3``(5pi)/6`(d) `pi/3`A. `(pi)/6`B. `(2pi)/3`C. `(5pi)/6`D. `(pi)/3` |
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Answer» Correct Answer - D `I=int_(5//2) ^(5)(sqrt((25-x^(2))^(3)))/(x^(4)) dx` Let `x=5 sin theta` `:.dx=5 cos theta d theta` `:. I=int_(pi//6)^(pi//2)(sqrt((25-25sin^(2) theta)^(3)))/(5^(4)sin^(4) theta)` `=int_(pi//6)^(pi//2) (5^(3)cos ^(3) theta. 5 cos theta)/(5^(4)sin^(4) theta) d theta` `=int_(pi//6)^(pi//2) cot^(2) theta (cosec^(2) theta-1)d theta` `=int_(pi//6)^(pi//2) cot^(2) theta cosec^(2) theta d theta -int_(pi//6)^(pi//2) cot^(2) theta d theta` `=int_(pi//6)^(pi//2) cot^(2) theta cosec^(2) theta d theta -int_(pi//6)^(pi//2) (cosec^(2) theta -1)d theta` `=[-(cot^(3) theta)/3+cot theta + theta]_(pi//6)^(pi//2)` `=-0+0+(pi)/2-(-(3sqrt(3))/3+sqrt(3)+(pi)/6)=(pi)/3` |
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