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Mathematics Question on Integrals of Some Particular Functions

∫−1117x5−x4+29x3−31x+1x2+1dx\int_{-1}^{1}{\frac{17{{x}^{5}}-{{x}^{4}}+29{{x}^{3}}-31x+1}{{{x}^{2}}+1}}dx is

A

44656

B

44685

C

44654

D

44624

Answer

44654

Explanation

Solution

Let I=∫−1117x5−x4+28x3−31x+1x2+1dxI=\int_{-1}^{1}{\frac{17{{x}^{5}}-{{x}^{4}}+28{{x}^{3}}-31x+1}{{{x}^{2}}+1}}dx I=∫−1117x5+29x3−31xx2+1dx−∫−11x4−1x2+1dxI=\int_{-1}^{1}{\frac{17{{x}^{5}}+29{{x}^{3}}-31x}{{{x}^{2}}+1}}dx-\int_{-1}^{1}{\frac{{{x}^{4}}-1}{{{x}^{2}}+1}}dx
=−2∫01(x2−1)(x2+1)(x2+1)dx=-2\int_{0}^{1}{\frac{({{x}^{2}}-1)({{x}^{2}}+1)}{({{x}^{2}}+1)}}dx
=2[(x33−x)]01=−2[13−1]=2\left[ \left( \frac{{{x}^{3}}}{3}-x \right) \right]_{0}^{1}=-2\left[ \frac{1}{3}-1 \right]
=43=\frac{4}{3}