第2009题：真分式化成部分分式之和
计算不定积分 ∫dx81−x4\int \dfrac{dx}{81-x^4}∫81−x4dx .
A. 136ln∣3+x3−x∣ \dfrac{1}{36} \ln \Big | \dfrac{3+x}{3-x} \Big | 361ln∣∣∣3−x3+x∣∣∣ +118arctanx3+C+\dfrac{1}{18} \arctan \dfrac{x}{3} +C+181arctan3x+C
B. 172ln∣3+x3−x∣\dfrac{1}{72} \ln \Big | \dfrac{3+x}{3-x} \Big |721ln∣∣∣3−x3+x∣∣∣ +136arctanx3+C+\dfrac{1}{36} \arctan \dfrac{x}{3} +C+361arctan3x+C
C. 154ln∣3+x3−x∣\dfrac{1}{54} \ln \Big | \dfrac{3+x}{3-x} \Big | 541ln∣∣∣3−x3+x∣∣∣ +127arctanx3+C+\dfrac{1}{27} \arctan \dfrac{x}{3} +C+271arctan3x+C
D. 1108ln∣3+x3−x∣ \dfrac{1}{108} \ln \Big | \dfrac{3+x}{3-x} \Big |1081ln∣∣∣3−x3+x∣∣∣ +154arctanx3+C+\dfrac{1}{54} \arctan \dfrac{x}{3} +C+541arctan3x+C
