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Evaluate:`int_(-pi/2)^(pi/2)sin^2xcos^2x(sinx+cosx)dx` |
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Answer» Correct Answer - `4//15` Let `=int_(-pi//2)^(pi//2)sin^(2)x cos^(2)x(sinx+cosx)dx` `=int_(-pi//2)^(pi//2) sin^(3)x cos^(2)x dx+int_(-pi//2)^(pi//2) sin^(2) x cos^(3)x dx`………….1 Since `sin^(3)x cos^(2)x` is an odd functional and `sin^(2)x cos^(3)x` is an even function. Therefore `int_(-pi//2)^(pi//2) sin^(3)x cos^(2)x dx=0` and `int_(-pi//2)^(pi//2) sin^(2) x cos^(3)x dx =int_(0)^(pi//2) sin^(2)x cos^(3)x dx`. `:. I=2int_(0)^(pi//2) sin^(2)x cos^(3)x dx` `=2int_(0)^(1)t^(2)(1-t^(2))dt` `=int_(0)^(1)(t^(2)-t^(4))dt=2[1/3-1/5]=4/15` |
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