e^xdx
To integrate the function e^x dx, you can use the technique of integration by parts
To integrate the function e^x dx, you can use the technique of integration by parts.
Integration by parts is a method that helps to simplify the integration of products of functions. It uses the formula:
∫u dv = uv – ∫v du
Here, we will take u = e^x and dv = dx.
To find du, we will differentiate u with respect to x:
du/dx = d/dx(e^x) = e^x
And to find v, we will integrate dv with respect to x:
∫dv = ∫dx = x
Now we can apply the formula of integration by parts:
∫e^x dx = ∫u dv = uv – ∫v du
= e^x * x – ∫x * e^x dx
We can solve the remaining integral ∫x * e^x dx using integration by parts again.
Let u = x and dv = e^x dx.
Then, du = dx and v = ∫dv = ∫e^x dx = e^x
Applying the formula of integration by parts again:
∫x * e^x dx = ∫u dv = uv – ∫v du
= x * e^x – ∫e^x dx
= x * e^x – e^x + C
where C is the constant of integration.
Therefore, the integral of e^x dx is x * e^x – e^x + C.
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