求不定积分:如图

令x = a*siny，dx = a*cosy dy，(a²-x²)^(3/2) = (a²-a²*sin²y)^(3/2) = (a²cos²y)^(3/2) = a³cos³y
tany = x / √(a²-x²)，cosy = √(a²-x²) / a
∫ x^4 / (a² - x²)^(3/2) dx
= ∫ a^4*sin^4y / (a³cos³y) * a*cosy dy
= a²∫ sin^4y / cos²y dy
= a²∫ sin²y*tan²y dy
= a²∫ sin²y*(sec²y - 1) dy
= a²∫ tan²y dy - a²∫ sin²y dy
= a²∫ (sec²y - 1) dy - (a²/2)∫ (1 - cos2x) dy
= a²(tany - y) - (a²/2)(y - 1/2*sin2y) + C
= a²x/√(a²-x²) - a²*arcsin(x/a) - (a²/2)*arcsin(x/a) + (a²/2)*(x/a)*√(a²-x²) / a + C
= a²x/√(a²-x²) - (3a²/2)arcsin(x/a) + (x/2)√(a²-x²) + C