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With the new courses yet to start, hopefully providing fresh material for new posts, I have been spending time going through some excercises from new textbooks.

As integrals have always been my favourite part of calculus, I decided to take down this solution, because it just looks nice. It also illustrates the principle: don't make a substitution, until it becomes obvious.

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Since we have and , so we need to choose negative sign when taking square root of the quadratic term.

It's now that the substitute becomes an obvious choice.

Some integrals yield only one type of substitution that really brings them into a convenient form. Any other method would make them more complicated. However, in some cases totally different methods can be applied with equal effect. In case of definite integrals it is of course not necessary to come back to original variable which makes things even easier. Here is one example

The most natural way is to apply a trigonometric substitute. We will not consider this method here. Instead an algebraic trick can be employed:

Alternatively we can use integration by parts:

Or apply an even more exotic treatment:

let

let

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For

and

,

the same result is obtained