M K.

asked • 06/23/16

Definite Integral with U-Substitution

I have an integral with ∫ (b=0, a=-1) and the inside function is (2t)/[(3+t2)3]dt. The directions tell me to "Evaluate the definite integral by making a u-substitution and integrating from u(a) to u(b). How do I go about this? 
 
Please include as much work and detail as you are able to. Thank you for your help in advance! 
 
~MK

1 Expert Answer

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Gregg O. answered • 06/23/16

For 3 semesters in college, top of my class in Calculus

M K.

Oh yeah, I definitely see how to integrate it! Sorry, I should have clarified that I'm really just not how to do the "integrating from u(a) to u(b)" part. Would you (or anyone else can, too, it'd be great) mind explaining how to do that? I was thinking the Fundamental Theorem of Calc, but I couldn't apply it correctly. Steps would so so helpful. Thank you!
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06/23/16

Gregg O.

When making a u-sub in a definite integral, it's common to leave things in terms of u after integrating.  In order to do this, the upper and lower limits are re-written.  For example, say the lower and upper limits of integration are 1 and 3, respectively.
 
Given our u = 3 + t2, the new lower limit becomes 3 + (1)2 = 4, while the new upper limit becomes 3 + (3)2 = 12.  What we're really doing is figuring out what u is equal to when t is equal to the upper and lower limits, and substituting these new values in place of the old limits of integration.
 
Use your actual values of a and b when solving the problem, and remember that unlike indefinite integrals with u-subs, you don't return to an expression in x before plugging in the upper and lower limits when changing the limits of integration to u(a) and u(b).
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06/23/16

M K.

Thanks a lot! This was really helpful!
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06/24/16

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