Double click anywhere & add an idea INTEGRATION by PARTS Why do we need to use integration by parts? Some integrals can't be solved with the integration techniques that we know. So we have to use integration by parts to solve them. And we use this formula to solve the integral. How do you use the formula? First, you have to figure out which part of the integral to use for "u" and which to use for "dv." To do so, you have to use the pneumonic "LIATE." The L stands for logarithmic

The I stands for inverse trigonometric

The A stands for algebraic

The T stands for trigonometric &

The E stands for exponential And which ever part of the integral comes first in the pneumonic is your "u" and the one left behind is your "dv." You still have to find your "du" and "v." To find your "du", you have to take the derivative of "u".

Then, to find "v," you have to integrate "dv," but don't add the constant. Now let's try one together Our "u" should be "x" and our "dv" should be e^x. Why? Because "x" is algebraic, which means it has to be "u", since it comes before the exponential part of the integral, e^x, when you follow the pneumonic "LIATE." So let's find our "du" by taking the derivative of "x" and our "v" by integrating "e^x." We should have gotten "dx" for our "du" and "e^x" for "v". Now that you have your parts, all you have to do is put them together using the formula. - = Then we have to integrate the second integral and the answer is "e^x+C." So, the integration of using integration by parts is: = - There are some integrals, when using integration by parts, that need even more steps to find the answer Here's an example: Now try one on your own You should have gotten: Guess what? There's a shortcut! *Unfortunately, you can only use the shortcut if your "u" goes to zero when you keep taking the derivative. The shortcut is called the "Tabular Method."

And it looks something like this: To use the Tabular Method, you have to take the derivative of the left side beginning with your "u" until you get to zero, and you have to integrate the right side beginning with your "dv" until you have something sitting next to your zero. Then, you have to alternate the sign in front of each derivative starting with a positive sign in front of "u". After that, you would multiply "u" with the second integrand ("dv" would be the first), the first derivative with the third integrand, the second derivative with the fourth integrand, and so on. Here's another example In this example, the "dv" part was "dx" because it could be integrated. Try this one Did you get this: You probably won't find integration by parts in the free response section of the ap exam, but it will be in the multiple choice section. Dr. Calculus THIS by zianga abraham

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# Integration by Parts

AP Calculus BC

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