InterviewSolution
Saved Bookmarks
InterviewSolution
| 1. |
`int _(0)^(a) (x^(4)dx)/((a^(2)+x^(2))^(4))` is equal toA. `(1)/(16a^(3))((pi)/(4)-(1)/(3))`B. `(1)/(16a^(3))((pi)/(4)+(1)/(3))`C. `(1)/(16)a^(3)((pi)/(4)-(1)/(3))`D. `(1)/(16)a^(3)((pi)/(4)+(1)/(3))` |
|
Answer» Correct Answer - A Put ` x= a tan theta rArr dx = a sec^(2) theta d theta ` ` :. int _(0)^(a) (x^(4) dx)/((a^(2)+x^(2))) = int _(0)^(pi//4) (a^(4) tan^(4) theta . a sec^(2) theta d theta )/(a^(8) sec^(8) theta )` ` = 1/(a^(3)) int _(0)^(pi//4) sin^(4) cos^(2) theta d theta ` ` = 1/(a^(3)) = int _(0)^(pi//4) [ ((1-cos2theta)^(2))/4 - ((1-cos 2 theta )^(3))/8 ] d theta ` ` = 1/(8a^(3)) int _(0)^(pi//4) (1+ cos 2theta ) (1+ cos^(2) 2theta - 2 cos 2 theta ) d theta ` ` = 1/(8a^(3)) int _(0)^(pi//4) (1- cos 2 theta - cos^(2) 2theta + cos^(3) 2 theta ) d theta ` ` = 1/(32a^(3)) int _(0)^(pi//4) (2 - cos 2theta - cos 4 theta + cos 6theta ) d theta` ` = 1/ (32a^(3)) [ 2theta - (sin2theta )/2 - (sin 4theta )/2 + (sin 6 theta )/6]_(0)^(pi//4)` ` =1/(16a^(3)) (pi/4 - 1/3)` |
|