Calculus (integration)

[errant mason]

How would I do part A?

[cyan hamlet]

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[torpid bone]

Try expressing G(x) in terms of F(t) first.

[errant mason]

does that mean to just replace e^t^2 with F(t)?

would I split up the integral?

[torpid bone]

No, F(t) is the antiderivative of e^(t^2).

[errant mason]

Is that what it means to be a primitive?

[torpid bone]

Yes. That term just isn't used as often now.

Using F(t), you can express G(x) using the fundamental theorem of calculus.

[errant mason]

Sorry I'm a little lost here, perhaps due to terminology, an antiderivative is the same as taking the integral right?

[torpid bone]

The indefinite integral of f(x) is ∫(f(x)dx) = F(x) + C, where F(x) is the antiderivative of f(x).
By the fundamental theorem of calculus, the definite integral can be calculated as ∫(f(x)dx, a, b) = F(b) - F(a).

[errant mason]

I think i understand! Thank you

[fathom violet]

@errant mason has given 1 rep to @torpid bone

[torpid bone]

Great! You're welcome 😁

[errant mason]

Have I done it correctly?

Just wanted to check i've understood properly

[torpid bone]

No. The derivative of F(x) is f(x), but the derivative of, say, F(3x) is not f(3x).

You forgot about the chain rule.

[errant mason]

ahha, thank you

[fathom violet]

@errant mason has given 1 rep to @torpid bone

[errant mason]

So the derivative of F(3x) = 3f(3x)

[torpid bone]

Yes.

[errant mason]

Tysm I don't know why it's taking my brain a long time to get around this

[torpid bone]

No worries! Take your time.

[errant mason]

so it would be instead
3e^(9x^2) + 2e^(4x^2)

[torpid bone]

Minus, but yeah.

[errant mason]

Whoops yup minus