8.6.4 Beyond Integration by Parts

We started this section with an example of a CAS integration that looks like it must have something to do with integration by parts:

∫ e x ⁢ sin ( x ) d ⁢ x = e x 2 ⁢ [ sin ( x ) - cos ( x ) ] .

Indeed, it does have something to do with integration by parts, as we see in the next Activity, but the connection is not a simple one.

Activity 3

1. Use the integration by parts formula to transform \(\displaystyle\int\,e^x\,\sin(x)\,dx\) by taking \(u=e^x\) and \(dv=\sin(x)\,dx\).
2. Use the integration by parts formula to transform \(\displaystyle\int\,e^x\,\sin(x)\,dx\) by taking \(u=\sin(x)\) and \(dv=e^x\,dx\).
3. Parts a. and b. produce different transformations, but they leave you with the same integral to be evaluated, one that is neither "simpler" nor "more complicated" than the problem we started with. Whichever you prefer of a. or b., use the same strategy to transform the remaining integral.
4. It appears from the result of part c. that we're getting nowhere — that we're right back where we started from. Don't give up too easily! In the equation \(\displaystyle\int\,e^x\,\sin(x)\,dx = uv-\int\,v\,du\) carefully substitute your choice of \(u\), \(v\), and \(du\) from part a. or b. Then do the same in the remaining integral, using your result from part c. (Be especially careful with the minus signs.) Now you should be able to solve the equation for the integral we wanted to evaluate. Do it!

Comment on Activity 3

Now we will see what a computer algebra system will do with this problem. First, click on the Wolfram|Alpha icon for the free online CAS service, and then click on "Step-by-step solution" to see how this CAS gets its answer. The result is not remarkably enlightening. Instead of following your steps in Activity 3, this CAS appears to have "memorized" a more general formula and just plugged in the constants. That can work for a computer system with a large, infallible memory, but it won't work for you.

Checkpoint 3

Activity 4

We repeat here Exercise 3 from section 8.4:

Use integration by parts to evaluate each of the following indefinite integrals. In each case, think about what gets simpler when you differentiate one of the parts.

a.   \(\displaystyle\int\;t\,\sin\,2t\,dt\)	b.   \(\displaystyle\int\;t^2\,\sin\,2t\,dt\)	c.   \(\displaystyle\int\;t^3\,\sin\,2t\,dt\)
d.   \(\displaystyle\int\,\ln\,t \,dt\)  (Hint: Consider \(\ln\,t\) as \(1 \times \ln\,t\).)
e.   \(\displaystyle\int\,t\,\ln\,t\,dt\)	f.   \(\displaystyle\int\,\sin^{-1} t\,dt\)	g.   \(\displaystyle\int\,t\,\sin^{-1} t \;dt\)

You may have already evaluated some or all of these integrals. Start this activity by writing down likely choices for \(u\) and \(dv\) as a first step in integration by parts for each of the seven integrals. Then take note of connections between two or more of the problems that suggest the answer to one may be used as a step in another. Finally, use the Indefinite Integral tool to evaluate all seven integrals and check the results by differentiation. Take note of any result that seems surprising.

Comment on Activity 4

Note 2 – Checking