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The value of the definite integral `int_(-1)^(1)(1+x)^(1//2)(1-x)^(3//2)dx` equalsA. `pi`B. `(3pi)/4`C. `(pi)/4`D. `(pi)/2` |
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Answer» Correct Answer - D `I=int_(-1)^(1)(1+x)^(1//2)(1-x)^(3//2)dx` `:.I=int_(-1)^(1)(1-x)^(1//2)(1+x)^(3//2)dx` (Replacing `x` by` -x`) Adding we get `2I=int_(-1)^(1)(1+x)^(1//2)(1-x)^(1//2)[(1-x)+(1+x)]dx` or `I=int_(-1)^(1)sqrt(1-x^(2))dx` `:. I=2int_(0)^(1)sqrt(1-x^(2))dx` `=2[x/2sqrt(1-x^(2)+1/2sin^(-1)x)]_(0)^(1)` `=2[0+1/2 . (pi)/2-0-0]=(pi)/2` |
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