In this discussion, we are going to evaluate integrals using integration by parts.
The formula for Integration by Parts is as follows:
The idea is to choose a value for u and dv such that, when the above formula is applied, the integral is easier to evaluate.
In some cases, integration by parts may have to be used more than once.
Below are the three examples that will be solved, showing all steps:
Be sure to read this discussion until the very end because, after working a few examples, an explanation will be given for how to make the best choice for u and dv.
Example 1
Note that “+ C” must always be included in the answer after evaluating indefinite integrals.
For the sake of convenience, “+ C” will only be included in the final answer after all steps have been simplified.
Example 2
For definite integrals (i.e., integrals that have upper and lower limits), there is no need to include “+ C”.
Example 3
Note:
It is also correct to choose
In this case, the answer would be
Although the graphs of
do not lay on top of each other, they are, in fact, the same graph that differ only by a vertical shift of “C”.
Thus, by including “+ C” for both answers, the value of C accounts for any vertical shifts that will place the graphs directly on top of each other.
If a definite integral was given, then the value of C would be found in both cases; and thus, both graphs would lay on top of each other.
How to Choose the Correct Value for u and dv
Regular Cases
In regular cases, problems like example 1 are given.
In this case, choose the value for u such that, after taking its derivative over and over, the value will eventually reach 0.
Choose dv such that, after integration over and over, the value will return to the original value of dv.
Special Case 1
In special cases, problems like example 2, where a log function is part of the integrand, are given.
When a log function is part of the integrand, dismiss any rules previously stated and let u = lnx and let the other function equal dv.
Special Case 2
Like example 3, if you see two functions such that, if you integrate or differentiate repeatedly, you eventually return to the original function, dismiss any rules previously stated and choose either function you wish for the value of u.
Then use integration by parts until you reach a repeated integral (the one you started with) and solve for that integral.
Examples of repeated integrals are sine, cosine, and ex.
Best wishes on your math journey!
✚MathAngel369✚
Angela Morgan