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- - **Definite integration by parts with sub**
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definite integration by parts with subhello, i am stuck on how to do this
I know how to do it for an indefinite integral, but it gets confusing for a definite integral. from my knowledge, when doing a definite integral, you have to change the upper and lower limit. but when it comes to integration by parts for a definite integral using substitution i get completely lost. my prof made an example using a indefinite integral, but not a definite integral. is there a methodology to solving these? |

Re: definite integration by parts with subQuote:
Quote:
Here's an example that is done both ways, using an ordinary substitution: [tex]\int_1^2 2x(x^2 + 1)^3 dx[/tex] 1. Limits of integration unchanged u = x ^{2} + 1, du = 2xdx[tex]\int_1^2 2x(x^2 + 1)^3 dx = \int_{x = 1}^2 u^3 du = \left.\frac{u^4}{4}\right|_{x = 1}^2[/tex] [tex]= \left.\frac{(x^2 + 1)^4}{4}\right|_{x = 1}^2 = \frac{625}{4} - \frac{16}{4} = \frac{609}{4}[/tex] 2. Limits of integration changed per substitution u = x ^{2} + 1, du = 2xdx[tex]\int_1^2 2x(x^2 + 1)^3 dx = \int_{u = 2}^5 u^3 du = \left.\frac{u^4}{4}\right|_{u = 2}^5[/tex] [tex]= \frac{625}{4} - \frac{16}{4} = \frac{609}{4}[/tex] In #2, when x = 1, u = 1 ^{2} + 1 = 2,and when x = 2, u = 2 ^{2} + 1 = 5 |

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