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Integrating e^x and 1/x

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Let's focus on two special cases of integration, and . We will look at different forms of these functions and what the rules are for integrating these functions.

You might recall that the general formula for integrating is . However, for functions that are not in the form , things start to get a bit trickier. The integral of a function is the opposite of the differential, which means that if you take a function, differentiate and then integrate it, you should get back the original function. For example:

→ differentiate → → integrate →

For , we know that the derivative is also , and therefore the integral of is , where c is just the constant of integration.

Note that this works both ways so if you integrated a function first and then differentiated it, you would also get back the original function.

We know that after integrating we have to be able to differentiate the resulting function to get back to the original function. We also know that the integral of is . If we were to integrate , we could use a substitution . So and dx = 2du. Therefore,

As 2 is a constant we can take it outside the integral and simply integrate which is .

Now we simply substitute back in to get

You might remember that the derivative of ln x is . Therefore, the integral of is ln x + c.

If you want to integrate or any other number / x you simply multiply ln x by that number. So

You might have spotted that is a combination of and . However, integrating this function is more complex than combining these two integrals together. The integral of this function, actually cannot be determined exactly and is therefore referred to as EI (x) or the Exponential Integral of x. So we simply say that

- The integral of
- The integral of is ln x + c

The integral of e^x is e^x+c.

The integral of e^3x is 1/3e^3x+c

The integral of 1/x is lnx + c

More about Integrating e^x and 1/x

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