The integration by parts method is not straightforward. It requires some thought, and the student must make two initial choices. Successfully working the exercises demands these choices be wise ones. And it may be necessary to repeat the process. Students often think they’ve failed if one application doesn’t yield the solution.

## Sample Problem #1

Below is an example of a question where integration by parts is applied twice. Note that in the directions of the question we point out that it might be used more than once.

### Solution

In the Practice mode, students have access to learning aids to help them understand how to tackle each problem. For example, they can choose to view the solution to the problem in the **Tutor** area.

The solution to this problem clearly explains how and why we pick *u* and *dv* and shows all the steps that take place to get the final answer.

### Step-by-Step

If students want to try answering the problem, but they do not know where to start, they have access to **Step-by-Step**. Step-by-Step provides a step-by-step breakdown of the problem, walking the student through the problem in manageable pieces. While it provides plenty of guidance, the Step-by-Step portion does ask the student to input the results of each step so they are learning as they go.

Below is Step 1, which reminds the student that their choices for *u* and *dv *should be made with the goal of producing a simpler integral.

Once the choices for *u* and *dv* are made, in Step 2 the student needs to find *du* and *v*.

In Step 3 the given integral is rewritten based on the method of integration by parts, and the student is being prompted and guided that integration by parts needs to be applied again. Therefore, the student needs to determine *u* and *dv* for the new integral resulting from the first application of the integration by parts method.

Once the choices for *u* and *dv* for the new integral to be evaluated by integration by parts were made, in Step 4 the student needs to find *du* and *v*.

In Step 5 the intermediate integral is rewritten based on the method of integration by parts and the student is prompted to evaluate it.

In the last step, the student puts together all the pieces found in the previous steps to find the result of the given integral.

### Explain Error

Another helpful learning aid provided in Hawkes’ courseware is **Explain Error**, which gives students precise feedback from the system’s artificial intelligence. It anticipates and

diagnoses specific errors, stopping students in their tracks and showing them not only that their answer is incorrect, but *why* it is incorrect.

Let’s say the student forgets to use C for the constant of integration.

When the student selects Explain Error, they receive this detailed feedback:

After the student reads the explanation, they can go back into Practice to add the constant of integration C:

Now, when applying the integration by parts the second time, let’s say the student makes a mistake in the sign of the antiderivative of sin*t*. So, instead of having *v* = – cos *t*, the student writes *v* = cos *t*.

This sign mistake leads to the following incorrect answer and the corresponding explanation.

After the student reads the Explain Error explanation, they can go back into Practice to modify their answer.

## Sample Problem #2

Below is a new question, which can be solved by different methods: integration by parts or *u*-substitution.

If the student were to choose *u = x* and *dv = *ln(2*x*^{2})*dx*, then *v* is very difficult to find and the integral becomes more complicated. Therefore, the best choices in this case are *u = *ln(2*x*^{2}) and *dv = xdv*.

### Solution

Below is the thorough solution. Note that this question also can be solved by starting with *u*-substitution. Our solution first shows the method of integration by parts, then it shows the *u*-substitution method.

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