How to solve this: $\int_0^4 e^{(x-2)^4}~dx=K$, find the value of $\int_0^4 xe^{(x-2)^4}~dx$

Question

How to solve this: Assume $$\int_0^4 e^{(x-2)^4}~dx=K,$$ I need to find the value of $$\int_0^4 xe^{(x-2)^4}~dx$$

I fail to use the integration by parts. How else can I solve this?

Answer 1

We know that , $$\int_a^bf(x)dx = \int_a^bf(a+b-x)dx $$

Taking $a=0$ $b=4$ we have

$$\color{blue}{\int_0^4 (x-2)e^{(x-2)^4}~dx = \int_0^4 (4-x-2)e^{(4-x-2)^4}~dx =-\int_0^4 (x-2)e^{(x-2)^4}~dx}$$

Hence, $$\color{blue}{\int_0^4 (x-2)e^{(x-2)^4}~dx = 0}$$ Therefore, $$\int_0^4 xe^{(x-2)^4}~dx =2\int_0^4 e^{(x-2)^4}~dx+\int_0^4 (x-2)e^{(x-2)^4}~dx\\=2K+\int_0^4 (x-2)e^{(x-2)^4}~dx = 2K$$

$$\color{red}{\int_0^4 xe^{(x-2)^4}~dx = 2K}$$