Question

Help plz.!! Given that `y=e^(x^2)`, find `dy/dx` Hence find `int xe^(x^2)dx`?

Answer 1

Use the chain rule to find `dy/dx`.

`dy/dx =(2x)e^(x^2)`

As for the chain rule, let `u = x^2`. Then `du = 2xdx` and `dx = (du)/(2x)`.

`I = 1/2int e^udu`

`I = 1/2e^u + C`

`I = 1/2e^(x^2) + C`

Alternatively we could have said

`I = 1/2int (2x)e^(x^2)dx`

`I = 1/2e^(x^2) + C` (because integration is the opposite of differentiation)

This is the same answer we got using the substitution.

Hopefully this helps!