1. Try performing the following integral
You cannot do this (easily) with the techniques we’ve discussed so far. So lets introduce a new one: integration by parts.
2. We should recall the product rule for differentiation:
Now we apply the Fundamental Theorem of Calculus to both sides, obtaining
So the left hand side simplifies, and the equation (3) becomes
and simple algebraic manipulation gives us the final result
What happened to the constant of integration? Well, it is “absorbed” into the integrals we’ve yet to perform (i.e., don’t worry about it the other integrals will produce the constants of integration which we’ll keep track of).
Example 1. Returning to Equation (1), i.e., considering the problem
What to do? Well, let
(5) and .
This produces the correct expression for the integrand, namely
Thus we use the formula for integration by parts writing
Observe that the right hand side simplifies to
where we suppress the constant of integration (i.e., we throw it away).
Thus integration by parts solves Equation (1) quite simply as .
Exercise 1. Calculate ???
Exercise 2. Compute ???
Exercise 3. Evaluate .
Exercise 4. Consider where we abuse notation writing .
Exercise 5. Repeatedly use integration by parts several times and evaluate .
Exercise 6 [Math.SE]. Consider
What is ?
Exercise 7 [Math.SE]. Calculate
Exercise 8 [Math.SE]. Denote to be the function which returns the greatest integer smaller than or equal to . For example, and . Calculate
for an arbitrary positive integer .