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Evaluate the integral using substitution

Use the method of substitution to evaluate the following integral:

Let

Then, we can evaluate the integral,

Now, we make a second substitution. Let

Then, we continue evaluating the integral,

1. There’s an alternative solution to this question, namely by using the substitution , so we have and . Then, the integral . However, . So, .

2. Johny Diala says:

In my opinion, there are a few minor errors here. In the top of your solution, you have

It should be

Moreover, during the end, you assert

though this again presumes .

How do we deal with this? You can, firstly, only solve the integral for so your original substitution is one-to-one. Now, since the integrand is even, it follows that the primitive is odd, and hence we have a solution

for and

for . With this, one can simplify the integral to

• Johny Diala says:

After “it should be”, there was supposed to “plus-minus” symbol on the RHS. For some reason it did not render.