Question

求不定积分：（3x^4+2x^3+4x^2+2x+5)/(x^2+1)

Answer 1

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