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 to

Let f(x) and phi(x) are two continuous function on R satisfying phi(x)

1.	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 to
Answer» <P>`int_(p)^(q)g(X)dx` `(n-m)int_(0)^(ALPHA)g(x)dx` `int_(p)^(alpha)g(x)dx+(n-m)int_(0)^(alpha)g(2X)dx` `int_(p)^(q)g(x)dx+((n-m))/2int_(0)^(2ALPHA)g(x)dx` Solution :`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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