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Problem 29 Medium Difficulty

Use a power series to approximate the definite integral to six decimal places.
$ \int^{0.3}_0 \frac {x}{1 + x^3} dx $

Answer

0.044522

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Video Transcript

use a power Siri's to absolute mate. The definite in girl to six decimal places. All right, so Okay, well, first expand is to his power serious. And then Bruce Inmate, the baton in the world to six decimal places. Okay, so is it because? And they were from zero two zero one three in six times, Monets wanted our in extra or three in and from zero to infinity. Yes, And this is Uncle Chu. Okay, we can change Thie order of some and the Negro, and it becomes in the Girl of X to the three. M plus one, the eggs. So this is equal to of Missouri Divinity once went off in, sir, Ask me X to the power through in plus two o three, plus two. And from zero to zero point three. All right, so, yeah, we can only take the term. Let's say if he goes to Well, well, zero. So it's gonna be X to the power of two. So is X square over too have elsewhere. And in this one, it's become too. It becomes tio minus. Okay, so we plug in and Nikos one, it's extra power five oh, or five in, Plus hire older term, which is too small, which is too small for this difference, so can only calculate the first tutor. So it's gonna be half of square miners. One fifth Time's s for five. So minus zero, the plugging actually cost zero point three and this is equal Tio so half of zero point zero nine minus one fifty times. So, thereupon three times times, they're gonna authority. Power flow is equals to three times Tintin to power minus one to our five, which goes to Syria's syrup on zero for five months we're going to comes, so it's gonna be two. Thirty four to thirty to forty three times ten to the power of Amon. It's fine, and we can use a calculator to simply buy it. And the result? It's going to be just a moment says they're one to comes two, forty three times. So ten to horror oneness for syrup on zero zero zero zero one. Yeah, and the result is Okay, so it says to six decimal places. All right, so the realist syrup on zero four four five one four. Okay, that is thie approximation of the definite, definite integral to six decimal places.

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