Mastering Integration by Parts: Finding the Integral of e^x dx

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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