不定积分1⼀(x+(a^2+x^2)^1⼀2)

∫ 1/[x + √(a² + x²)] dx
= ∫ 1/[x + √(a² + x²)] * [x - √(a² + x²)]/[x - √(a² + x²)] dx，分母有理化
= ∫ [x - √(a² + x²)]/[x² - (a² + x²)] dx
= (- 1/a²)∫ [x - √(a² + x²)] dx
= (- 1/a²)∫ x dx + (1/a²)∫ √(a² + x²) dx，x = a*tanz，dx = a*sec²z dz
= (- 1/a²)(x²/2) + (1/a²)∫ |a*secz| * (a*sec²z dz)
= - x²/(2a²) + ∫ sec³z dz
= - x²/(2a²) + (1/2)secztanz + (1/2)ln|secz + tanz| + C
= - x²/(2a²) + (1/2)(x/a)[√(a² + x²)/a] + (1/2)ln|x/a + √(a² + x²)/a| + C
= - x²/(2a²) + [x/(2a²)]√(a² + x²) + (1/2)ln|x + √(a² + x²)| + C

ln(x + (a^2 + x^2)^(1/2))/2 - x^2/(2*a^2) + (x*(a^2 + x^2)^(1/2))/(2*a^2)

(2*atan(x/(a^2 - 1)^(1/2) + 1/(a^2 - 1)^(1/2)))/(a^2 - 1)^(1/2)+C