Question

If $m,n$ are even integers and $p,q\in R,$ then ${\int }_{p+m}^{q+n\alpha }g\left(t\right)dt$ equal to

• A. $a{\int }_p^{q}g\left(x\right)dx$
• B. $(n-m){\int }_0^{\alpha }g\left(x\right)dx$
• C. ${\int }_p^{q}g\left(x\right)dx+(n-m){\int }_0^{\alpha }g\left(2x\right)dx$
• D. ${\int }_p^{q}g\left(x\right)dx+(n-m){\int }_0^{\alpha }g\left(x\right)dx$

Answer 1

The correct option is D ${\int }_p^{q}g\left(x\right)dx+(n-m){\int }_0^{\alpha }g\left(x\right)dx$

${\int }_{p+m}^{q+n\alpha }g\left(t\right)dt={\int }_{p+m}^{p}g\left(x\right)dx+{\int }_p^{q}g\left(x\right)dx+{\int }_q^{q+n\alpha }g\left(x\right)dx$
$=-m{\int }_0^{\alpha }g\left(x\right)dx+{\int }_p^{q}g\left(x\right)dx+n{\int }_0^{\alpha }g\left(x\right)dx$
$={\int }_p^{q}g\left(x\right)dx+(n-m){\int }_0^{\alpha }g\left(x\right)dx$