Integral by Tricky Substitution

This is an integral that can be solved using substitution. However, the substitution is not so obvious. That is why I decided to put this integral as a solved example here:

At first sight we can't see how substitution will work here. However, we can try it anyway. We'll make:

However, in the numerator we don't have an x, we have x to the fifth power. We can write it, though, like this:

And from our substitution we have that:

Replacing that in our integral we have:

Integrating these simple integrals we get:

Substituting back our u, we have:

You may check that this is indeed the indefinite integral of the function by taking the derivative. So, this was an interesting example of the substitution method. It is different in that we needed to work a bit more algebraically to make the substitution.

It is very nice to know you can do this.

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