Evaluating Indefinite Integrals for Dan

danago · Oct 25, 2006

Hey. Say i was given this indefinite integral to evaluate:

http://img126.imageshack.us/img126/6374/aaaaog3.gif

How could i do that? I can do it by first expanding it all, but that takes a very long time and is quite tedious, especially with such a large index as 7. Is there another way i can do that?

Thanks,
Dan.

StatusX · Oct 25, 2006

You could use substitution.

danago · Oct 25, 2006

What would i substitute out? I tried using substitution, but couldn't get very far.

danago · Oct 25, 2006

http://img109.imageshack.us/img109/8527/aaaaac7.gif

Where from there?

HallsofIvy · Oct 25, 2006

Why squared? that leaves a fraction power. I would just let
u= 3x+2 then du= 3dx so dx= (1/3)du. Also, x= u/3+ 2/3 so
x(3x+2)⁷dx= (u/3+ 2/3)(u⁷)(1/3du)= (1/9)(u⁸+ 2u⁷)dx

Danago, you have remember to replace dx with du. If u= (3x+2)², then du= 2(3x+2)dx= (6x+ 4)dx. I think courtrigrad's point was that you can use that "x" in the integeral to help with that. But with that "4dx" still left, I think u= 3x+2 is simpler.

courtrigrad · Oct 25, 2006

doesn't x = u/3 - 2/3

danago · Oct 25, 2006

HallsofIvy said:

Why squared? that leaves a fraction power. I would just let
u= 3x+2 then du= 3dx so dx= (1/3)du. Also, x= u/3+ 2/3 so
x(3x+2)⁷dx= (u/3+ 2/3)(u⁷)(1/3du)= (1/9)(u⁸+ 2u⁷)dx

Danago, you have remember to replace dx with du. If u= (3x+2)², then du= 2(3x+2)dx= (6x+ 4)dx. I think courtrigrad's point was that you can use that "x" in the integeral to help with that. But with that "4dx" still left, I think u= 3x+2 is simpler.

Hmmm...im a bit lost. I understand up to "dx= (1/3)du", but where does the "x= u/3+ 2/3" come from? If it is anything more advanced than substitution, i think ill leave it there, because i just made this question up out of curiosity, not because i need to know it for school, and even substitution is more advanced than what we've been doing at school, but i learned it to make some of the question we do simpler.

courtrigrad · Oct 25, 2006

If u = 3x +2, then x = u/3 - 2/3. You have to solve for x.

danago · Oct 25, 2006

oh lol. That simple :P

danago · Oct 25, 2006

Is this right?
http://img150.imageshack.us/img150/1972/aaaaka4.gif

Max Eilerson · Oct 25, 2006

Yes it's correct, but you can tidy it up a little.

danago · Oct 25, 2006

ok thanks everyone :)

Evaluating Indefinite Integrals for Dan

FAQ: Evaluating Indefinite Integrals for Dan

What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the anti-derivative of a function. It is a function that, when differentiated, gives the original function as its result.

Why is it important to evaluate indefinite integrals?

Evaluating indefinite integrals is important in order to find the exact value of a function at a given point. It also allows for the calculation of areas under curves and can be useful in solving differential equations.

What are the different methods for evaluating indefinite integrals?

There are several methods for evaluating indefinite integrals, including the power rule, substitution, integration by parts, and trigonometric substitution. Each method is useful for different types of functions and can be used to solve a variety of integration problems.

What is the role of limits in evaluating indefinite integrals?

Limits play a crucial role in evaluating indefinite integrals as they determine the bounds of integration. These bounds can affect the final value of the integral and must be carefully chosen to accurately represent the area under the curve.

How can I check if my solution to an indefinite integral is correct?

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