求不定积分：（3x^4+2x^3+4x^2+2x+5)/(x^2+1)

题目

求不定积分：（3x^4+2x^3+4x^2+2x+5)/(x^2+1)
x^3+x^2+x+4arctanx+c

答案

3x⁴ = 3x²[(x² + 1) - 1] = 3x²(x² + 1) - 3[(x² + 1) - 1] = 3x²(x² + 1) - 3(x² + 1) + 3
2x³ = 2x[(x² + 1) - 1] = 2x(x² + 1) - 2x
4x² = 4[(x² + 1) - 1] = 4(x² + 1) - 4
∴
3x⁴ + 2x³ + 4x² + 2x + 5
= [3x²(x² + 1) - 3(x² + 1) + 3] + [2x(x² + 1) - 2x] + [4(x² + 1) - 4] + 2x + 5
= (x² + 1)(3x² - 3 + 2x + 4) + (3 - 4 + 5)
= (x² + 1)(3x² + 2x + 1) + 4
∴
∫ [3x⁴ + 2x³ + 4x² + 2x + 5]/(x² + 1) dx
= [(x² + 1)(3x² + 2x + 1) + 4]/(x² + 1) dx
= ∫ (3x² + 2x +1) dx + 4∫ dx/(x² + 1)
= x³ + x² + x + 4arctan(x) + C