A curve has gradient e(power 4x)+e(power -x) at the point (X,Y). Given that the curve passes through the point (0,3), find the equation of the curve.
Plz Help!
The gradient function is $\displaystyle \displaystyle e^{4x} + e^{-x} $.
If $\displaystyle \displaystyle \frac{dy}{dx} = e^{4x} + e^{-x}$, then what is $\displaystyle \displaystyle y$? In other words, how do you undo taking a derivative?
The integration in question uses $\displaystyle \int e^{ax} = \dfrac{1}{a}e^{ax} +C$. If you don't know why this is the case you'll need to use a textbook or ask your teacher for some more tuition and/or examples on integration
Well, the most basic thing about integration is that it is, up to a constant, the inverse of differentiation. Let's say I take the function f(x) = sin(x), and I differentiate it to obtain f'(x) = cos(x). Well now, I can integrate it to get back to my original function:
$\displaystyle \int\cos(x)\,dx=\sin(x)+C.$
Notice the constant there. So use the antiderivative that e^(i*pi) just gave you, plus this fact, and then use the constant of integration to make your curve go through the desired point. Does that make sense?