Integration by Substitution | Solving the Integral of e^u with Respect to u

∫(e^u)du

To solve the integral of e^u with respect to u, we can apply a basic integration technique known as integration by substitution

To solve the integral of e^u with respect to u, we can apply a basic integration technique known as integration by substitution.

Let’s denote the integral of e^u with respect to u as I. We can rewrite the integral as:

∫(e^u)du = I

Now, let’s choose a substitution that simplifies the integral. In this case, we can set u equal to a new variable, let’s say v:

u = v

Differentiating both sides of this equation with respect to v, we have:

du/dv = 1

Solving for du, we get:

du = dv

Now, we substitute back into the original integral to express everything in terms of v:

I = ∫(e^v)du

Notice that we now have an integral in terms of v instead of u. But since du = dv, we can substitute this into the integral as well:

I = ∫(e^v)dv

At this point, we can easily evaluate the integral. The integral of e^v with respect to v is simply e^v plus a constant of integration. Therefore:

I = e^v + C

Finally, we substitute back the original variable u for v:

I = e^u + C

So, the integral of e^u with respect to u is e^u plus a constant of integration (C).

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