What is the general method for integrating by parts?

Answer 1

#intu# #dv=uv-int# #v# #du#

The trick with doing integration by parts comes in choosing your #u# and #dv# accordingly. If you choose poorly, the problem will usually become harder.
For example, consider #intxsinx# #dx#.
If you let #u=sinx# and #dv=x# #dx# (and accordingly #du=cosx# #dx# and #v=1/2x^2#) you will end up with
#intxsinx# #dx=1/2x^2sinx-1/2intx^2cosx# #dx#.
You can try to evaluate the integral #intx^2cosx# #dx# but if you make similar #u# and #dv# choices this problem will continue to get more complicated.
Instead, for #intxsinx# #dx#, let #u=x# and #dv=sinx# #dx# (and accordingly #du=1# #dx# and #v=-cosx#). Then you get
#intxsinx# #dx=-xcosx-int-cosx*(1)# #dx#
#=-xcosx+intcosx# #dx=-xcosx+sinx+C#.

This way, you can actually do the problem.

Now you are probably thinking "Is there a way to know to to make #u# and #dv# so I don't have to go through this process of trial and error?"
The answer is yes. Use the acronym LIPET to remember the order of choosing #u#, in order of best to choose to worst to choose.

BEST L: logarithmic stuff I: inverse trig stuff P: polynomialish stuff E: exponential stuff T: trig stuff WORST

Trial and error should help explain why this order is helpful.

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Answer 2

The general method for integrating by parts involves using the formula:

[ \int u , dv = uv - \int v , du ]

Where ( u ) and ( v ) are differentiable functions of ( x ).

To apply the method of integration by parts:

  1. Choose parts: Select ( u ) and ( dv ) such that differentiation of ( u ) or integration of ( dv ) simplifies the integral.

  2. Calculate differentials: Differentiate ( u ) to get ( du ) and integrate ( dv ) to get ( v ).

  3. Apply the formula: Substitute ( u ), ( dv ), ( du ), and ( v ) into the integration by parts formula.

  4. Evaluate the resulting integral: This may lead to a simpler integral, or it may require repeating the integration by parts method.

  5. Repeat if necessary: If the resulting integral is still not easy to evaluate, apply integration by parts again or use other integration techniques.

  6. Solve for the original integral: Once you have simplified the integral, solve for the original integral.

  7. Check for convergence: Ensure that the integral converges by checking the conditions for convergence if dealing with improper integrals.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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