InterviewSolution
Saved Bookmarks
| 1. |
If `I=int_(-pi//2)^(pi//2)(sin^(4)x)/(sin^(4)x+cos^(4)x)dx,` then the value of I is |
|
Answer» Correct Answer - C Let `f(x)=(sin^(4)x)/(sin^(4)x+cos^(4)x)` `f(-x)=(sin^(4)(-x))/(sin^(4)(-x)+cos^(4)(-x))` `=(sin^(4)(x))/(sin^(4)(x)+cos^(4)(x))=f(x)` `therefore` f is an even function. `thereforeint_(-pi//2)^(pi//2)(sin^(4)x)/(sin^(4)x+cos^(4)x)dx` `=2int_(0)^(pi//2)(sin^(4)x)/(sin^(4)x+cos^(4)x).dx` `thereforeI=2int_(0)^(pi//2)(sin^(4)x)/(sin^(4)x+cos^(4)x)dx` ... (i) By using the property, `int_(0)^(a)f(x)*dx=int_(0)^(a)f(a-x)*dx`, we get `I=2int_(0)^(pi//2)(sin^(4)(pi//2-x))/(sin^(4)(pi//2-x)+cos^(4)(pi//2)-x)dx` `=2int_(0)^(pi//2)(cos^(4)x)/(cos^(4)x+sin^(4)x)dx` . . . (ii) Adding equation (i) and(ii) ,we get `=2int_(0)^(pi//2)(sin^(4)x+cos^(4)x)/(sin^(4)x+cos^(4)x)dx` `I=int_(0)^(pi//2)dx` `I=[x]_(0)^(pi//2)=[(pi)/(2)-0]=(pi)/(2)` Hence ,the correct answer from the given alternative is |
|