How do you find the antiderivative of $e^(-2x)$ ?

Question

12567 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-02T10:11:37

$inte^(-2x)dx=-1/2e^(-2x)+C$

it is important to remember that

$d/dx(e^x)=e^x$

so let us see what happens if we differentiate the given function

$y=e^(-2x)$

$u=-2x=>(du)/(dx)=-2$

$y=e^u=>(dy)/(du)=e^u$

by the chain rule we have:

$(dy)/(dx)=(dy)/(du)xx(du)/(dx)$

giving us

$(dy)/(dx)=e^uxx-2e^(-2x)=-2e^(-2x)$

now integration is the reverse of differentiation, so comparing what we have after differentiating and the function we are given to integrate.

we have to adjust the function by a suitable constant to cancel the $" -2$

$inte^(-2x)dx=-1/2e^(-2x)+C$

if you now differentiate the resulting function you will see it gives the original function.

How do you find the antiderivative of e^(-2x)e^(-2x)?