Integration by Parts
The purpose of integration by parts is to replace a difficult integral with one that is easier to evaluate. The formula that allows us to do this is
Contents
Summary
Suppose we are trying to do the integration We notice that -substitution cannot be used, since neither nor is close to being the derivative of the other. A function which is the product of two different kinds of functions, like requires a new technique in order to be integrated, which is integration by parts. The rule is as follows:
This might look confusing at first, but it's actually very simple. Let's take a look at its proof and find out how easy and convenient it is:
Remember the product rule? It is the rule for differentiating the product of two functions, expressed in terms of
Integration by parts is exactly its antiderivative form. Integrating both sides gives
Let and Then we have a more compact expression:
Now that we know the rule, let's find the answer to the example above.
What is
Let and Then we have
where is the constant of integration.
Integration by Parts - Basic
As shown in the example above, we let one factor be and the other or Then our given problem will be and we can apply the rule of integration by parts. So then, what is the criterion to determine which of the factors will be Why can't we let and There is a rule to this too: Since is to be integrated, let be the easier to integrate. Easier means that the function changes less after integration. For instance, an exponential function, say is easy to integrate, since it doesn't change at all after integration. A trigonometric function will change to its counterpart. For instance, will change to Logarithmic functions are the most difficult to integrate. Here is an explicit demonstration of the rule:
The further to the right, the better to put as and the more on the left, the better to put as Let's see a few more examples.
Evaluate
has a pretty simple derivative, so let's say . Then , and , which implies.
where is the constant of integration. You can take the derivative to see that it is indeed our desired result.
Find the indefinite integral
According to the rule above, we should let and Then we have
where is the constant of integration.
Find the indefinite integral
According to the rule above, we should let and Then we have
We know what is from an example question above (otherwise, we would have to use integration by parts one more time). Therefore, we have
where is the constant of integration.
Notice that we needed to use integration by parts twice to solve this problem.
Find the indefinite integral
According to the rule above, we should let and Then we have
where is the constant of integration.
Find the indefinite integral
This one needs a trick. Think of as Then, according to the rule above, we should let and Then we have
where is the constant of integration.
This one will appear very frequently, so memorizing is highly recommended.