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Use a power series to approximate the definite integral to six decimal places.

$ \int^{1/2}_0 \arctan (x/2) dx $

$\approx 0.061865$

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Okay. Use the power series to approximated. Definitely grow to six decimal places. All right, we're gonna first expand. This function so equals two arc tangent. Half of X equals two. So head of Ike's minus have access to the power of three over three and plus some hierarchy. Term of X cube. So which is too small to be related in this? In this difference, the X and this is going to be Is this going to be, uh, X square over four, minus extra power four over 96. Last some higher on the term. So we can just even more. This part. Oh, and so it's from one half. So we're talking one half to dysfunction and minus these values at zero. Okay, Okay. Let me first check this, uh, is that if the correct So we take the derivative of it. So it's gonna be, uh, have axes, okay? And the rest of this is going to be four and six of 24 on the Terminator and execute on the numerator yet, as she's definitely correct. Okay. And we're gonna we're gonna evaluate this. This difference. We just plug in one half and it becomes to zero point 254 and 0.5 to power 4/96. So the final answer is zero point 061849 and we meet this requirements six decimal places. 12346 All right.

University of Illinois at Urbana-Champaign