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Let `f(x)` and `phi(x)` are two continuous function on `R` satisfying `phi(x)=int_(a)^(x)f(t)dt, a!=0` and another continuous function `g(x)` satisfying `g(x+alpha)+g(x)=0AA x epsilonR, alpha gt0`, and `int_(b)^(2k)g(t)dt` is independent of `b` If `m,n` are even integers and `p,q epsilon R`, then `int_(p+n alpha)^(q+n alpha)g(t)dt` is equal toA. `int_(p)^(q)g(x)dx`B. `(n-m)int_(0)^(alpha)g(x)dx`C. `int_(p)^(alpha)g(x)dx+(n-m)int_(0)^(alpha)g(2x)dx`D. `int_(p)^(q)g(x)dx+((n-m))/2int_(0)^(2alpha)g(x)dx` |
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Answer» Correct Answer - D `int_(p+m alpha)^(q+n alpha) g(t)dt=int_(p+m alpha)^(p)g(x)dx+int_(p)^(q)g(x)dx+int_(q)^(q+n alpha) g(x)dx` `=-m/2 int_(0)^(2 alpha) g(x)dx+int_(p)^(q)g(x)dx+n/2 int_(0)^(2alpha) g(x)dx` `=int_(p)^(q)g(x)dx+((n-m)/2)int_(0)^(2alpha) g(x)dx` |
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