Mathematics

For $$x\neq 2$$ , if $$ \int_{4-x}^x e^x(4-x)dx=2$$, then $$\int_{4-x}^x xe^{(4-x)}dx$$ is equal to


ANSWER

2


SOLUTION


$$\\\int_{4-x}^{x} e^x(4-x)dx=2\\let x=4-t\\then dx=-dt\\and if x=x, t=4-x\\if x=4-x, t=x\\\therefore\>I=\int_{x}^{4-x}e^{4-t}t(-dt)\\=\int_{4-x}^{x}e^{4-t}t dt=2\\replace \>t \>by x\\I=\int_{4-x}^{x}e^{4-x}x dx=2$$


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