Find the value of ' $k$ ', if it is given that $\int_0^{b-c} f(x+c) d x=k \int_c^b f(x) d x$

Question

Aicharya · Accepted Answer

$I=\int_0^{b-c} f(x+c) d x$ Put $x+c=t$, then at $x=0, t=c$ and at $x=b-c, t=b$ and $d x=d t$, so $I=\int_c^b f(t) d t=\int_c^b f(x) d x \Rightarrow \int_0^b f(x+c) d x=\int_c^b f(x) d x$ Therefore $k=1$

Find the value of ' \(k\) ', if it is given that \(\int_0^{b-c} f(x+c) d x=k \int_c^b f(x) d x\)

Solution