Question

$ \int_{-1}^{1}{\frac{17{{x}^{5}}-{{x}^{4}}+29{{x}^{3}}-31x+1}{{{x}^{2}}+1}}dx $ is

• A. 4/5
• B. 5/4
• C. 4/3
• D. 3/4
• E. 2/3

Answer 1

Answer: (C)

Let $ I=\int_{-1}^{1}{\frac{17{{x}^{5}}-{{x}^{4}}+28{{x}^{3}}-31x+1}{{{x}^{2}}+1}}dx $ $ I=\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\int_{0}^{1}{\frac{({{x}^{2}}-1)({{x}^{2}}+1)}{({{x}^{2}}+1)}}dx $
$=2\left[ \left( \frac{{{x}^{3}}}{3}-x \right) \right]_{0}^{1}=-2\left[ \frac{1}{3}-1 \right] $
$=\frac{4}{3} $