∫√(x²+a²)dx如何推导,

∫√(x²+a²)dx如何推导,
∫√(x²+a²)dx如何推导,

∫√(x²+a²)dx如何推导,
T = ∫√(x²+a²)dx
u = √(x²+a²), dv = dx, v = x
T = uv - ∫vdu = x√(x²+a²) - ∫xd√(x²+a²)
= x√(x²+a²) - ∫x²dx/√(x²+a²)
= x√(x²+a²) - ∫(x² + a² - a²)dx/√(x²+a²)
= x√(x²+a²) - ∫√(x²+a²)dx + a²∫dx/√(x²+a²)
= x√(x²+a²) - T + a²ln[x + √(x²+a²)] + c
2T = x√(x²+a²) + a²ln[x + √(x²+a²)] + c
T = ∫√(x²+a²)dx = (1/2)x√(x²+a²)+ (a²/2)ln[x + √(x²+a²)] + C