Antiderivative of 6e^x and 4x^2

[eclar56]

antiderivative of 6e^x and 4x^2

--> 6e^x
-->1/6e^x??


--> 4x^2
--> 2x^3??

 

[n.karthick]

Could you tell the steps you followed to find anti-derivative of $4x^2$ and how you got the answer.

 

[eclar56]

4x^2:
One of the teachers said today to add one to the power, and divide what's out front by what the power was. So 4/2 = 2, and 2 + 1 = 3..

 

[n.karthick]

Oh no! As far as adding one to the power is ok, but could you re-call correctly what has been told about the divisor.
You can refer to standard integrals in the PF library to know the integration of $x^n$.

 

[eclar56]

Ok I think I figured it out. Thanks heaps for helping anyway. Don't know what my teacher was talking about but! Still don't know the 6e^x one but..

 

[HallsofIvy]

The derivative of Cf(x), where C is a constant, is Cf'(x) so $(6e^x)'= 6(e^x)'$. Do you know what the derivative of $e^x$ is?

 

[eclar56]

It's just itself. So the whole thing is just the same? It's still just 6e^x..?

 

[Lancelot59]

Yes. Remember what happens when you derive something with a constant in front of it? You pull the constant out:

$$\frac{d}{dx} 6e^{x}$$
$$6\frac{d}{dx} e^{x}$$
$$6(e^{x})$$

The function e^xdoesn't change when you derive it unless you have something other than x in the exponent and have to use the chain rule.

 

[mg0stisha]

don't forget your constant of integration, but yes.

 

[Lancelot59]

don't forget your constant of integration, but yes.

My bad.

$$\int{6e^{x}} dx =6\int{e^{x}} dx = 6e^{x} + c$$