Integration with $e$

Question

What are the steps in this integration :

$$ \int4e^{t(u-4)}dt = \frac{-4}{u-4}e^{t(u-4)} $$

I've calculated simpler integration formulas such as $$ \int x^2dt = \frac{x^3}{3} $$

but I'm unsure how integral above is calculated as there is an $e$ involved.

Answer 1

$$I=\int4e^{(u-4)t}dt=4\int e^{(u-4)t}dt$$ Substitute $$w=(u-4)t\\\Rightarrow dw=(u-4)dt\\\Rightarrow \frac{dw}{u-4}=dt$$ Hence $$I=4\int e^{w}\frac{dw}{u-4}=\frac{4}{u-4}\int e^wdw$$ Since $\frac{d}{dx}e^x=e^x$, $$I=\frac{4}{u-4}e^w+C$$ Re-substitute: $$I=\frac{4}{u-4}e^{(u-4)t}+C$$

Answer 2

Since $\int e^t\,\mathrm dt=e^t$, $\int e^{at}\,\mathrm dt=\dfrac{e^{at}}a$.

Answer 3

Substituting $$m=t(u-4)$$ then you will get $$dm=(u-4)dt$$