Basic Integration Problem

Calculus Level 3

0 1 x 4 x 4 + 4 x 3 + 12 x 2 + 24 x + 24 d x \displaystyle \int_0^1 \dfrac{x^4}{x^4+4 x^3 + 12 x^2 + 24 x + 24} \: dx

Please round your answer to 8 decimal points.


The answer is 0.00366656.

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1 solution

Hai Bin Chang
Oct 25, 2017

0 1 x 4 x 4 + 4 x 3 + 12 x 2 + 24 x + 24 d x \displaystyle \int_0^1 \dfrac{x^4}{x^4 + 4 x^3 + 12 x^2 + 24 x + 24} \: dx = 0 1 ( 1 4 x 3 + 12 x 2 + 24 x + 24 x 4 + 4 x 3 + 12 x 2 + 24 x + 24 ) d x =\displaystyle \int_0^1 \left(1 - \dfrac{4x^3 + 12 x^2 + 24 x + 24}{x^4 + 4 x^3 + 12 x^2 + 24 x + 24}\right)\: dx = [ x ln ( x 4 + 4 x 3 + 12 x 2 + 24 x + 24 ) ] 0 1 = \displaystyle \left[x - \ln(x^4 + 4 x^3 + 12 x^2 + 24 x + 24)\right]_0^1 = 1 ln ( 65 ) + ln ( 24 ) 0.00366656 = 1 - \ln(65) + \ln(24) \approx 0.00366656

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