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Integration by Parts

This tutorial describes a technique for evaluating integrals using a powerful feature of Theorist, Integration by Parts. Integration by parts is used when your integrand consists of two parts multiplied together, and you know how to integrate one and how to differentiate the other. Since differentiating is usually easy, this technique is useful for integrals that can not be solved any other way. Integration by parts does not always solve your integral, since the end result includes another integral. The formula for the integration by parts technique shows this:

 

Enter the following integral by typing $(x*e^x *d*x):'p:2'p. Clarify to remove unnecessary parentheses.

Select the expression (click the integral sign or the palette button) then click the palette button (or choose Simplify from the Manipulate menu).

Theorist took the limits off the integral, but could not integrate it. This expression is a candidate for integration by parts. First we have to decide which part to integrate and which part to differentiate.

The ex stays the same if we integrate it or differentiate it. The x gets more complicated if we integrate it, but it turns into 1 if we differentiate it We should select the part to integrate, so we want to select ex and dx.

Try to make a multiple selection of ex and dx (as shown below) by dragging and holding the key.

Theorist does not allow these terms to be multiple selections because selecting the dx means selecting the product of all four things.

Therefore, we will use the Commute manipulation, by hand (with the or key). Grab the ex and move it to between the x and dx, like so:

This gives us:

. by Parts;

Choose Auto Casing from the Manipulate menu’s Preferences submenu (to turn it on). Select exdx and choose from the pop-up palette (or choose Int. by Parts from the ManipulateÁOther submenu).

 

(In your answer, the arbitrary constant c may have different subscripted numbers; this is fine.)

Select the expression to the right of the equals sign and click (or choose Expand from the Manipulate menu), which produces the final answer.

 

There is another way of separating ex and dx to integrate by parts, which we will do.

Copy the original integral expression to the clipboard. Make a new notebook and paste it in. Turn on Auto Casing. Select and Simplify the integral, as in the first step of the other method. Next, select dx alone and Simplify, which gives us a slightly different result.

Note the parentheses. Now the integral is the product of three things, the last of which is dx. Make a multiple selection of the ex and dx; this time Theorist allows it.

Give the Integrate by Parts command and Expand as before, to get the same final answer.

 

Try integrating º x2 cos(x) dx by parts. (Integrate the cos(x) dx part again.) Hint: you must do it twice.

Try integrating º x7 cos(x) dx by parts. It may be tedious.

Try integrating º ex cos(x) dx by integrating by parts on cos(x) dx. Hint: after doing it twice, use some algebra to solve for the integral.

Try the previous exercise, integrating exdx the second time. Integration by parts can lead you in circles if you are not careful.

Try integrating º ln(x) dx, but don’t cheat by simplifying it. Instead, integrate by parts just on the dx part. Although it does not seem like it should work, it does. Do it with Auto Simplify off to watch what happens step-by-step.

Find out if arbitrary constants interfere. Try doing an integration by parts with Auto Casing on.

 

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