Question

antidifferentiate : 2xe^(-x^2)

Answer 1

\[\int\limits_{}^{}2xe ^{-x ^{2}}\]

Answer 2

\[-e^{x^2}\]

Answer 3

how do u do it?

Answer 4

Well, it is an inmediate integral, because
\[\int g'(x)f(g(x))dx\]

Answer 5

\[\int\limits_{}^{}2xe ^{-x ^{2}} = -\int e^{-x^2}d(-x^2)\]

Answer 6

But you can try with a change of variable,
\[x^2=t\\2xdx=dt\]

Answer 7

so what happens if its g'x f(g(x)?

Answer 8

In this particular case, we have
\[\int g'(x)f'(g(x))dx=\int \frac{d(f(g(x)))}{dx} dx=f(g(x))\]

Answer 9

When you have integrals that cannot solve this way, but still are "inmediate", you can do a change of variable. I suggested x^2=t, but watching the answer of @wio you can also do, -x^2=t.
\[-x^2=t\\
-2xdx=dt\Rightarrow2xdx=-dt\\
\int 2xe^{x^2}dx=-\int e^tdt=-e^t=-e^{x^2}\]Don't forget to add the usual constant for indefinite integrals,
\[-e^{x^2}+K\]

Answer 10

Thankyou

Answer 11

;)