求根号下(a^2+x2)^3的不定积分

求根号下(a^2+x2)^3的不定积分

令x = a * tanz,dx = a * sec²z dz,sinz = x/√(a² + x²),cosz = a/√(a² + x²)
∫ √(a² + x²)³ dx
= ∫ (a² + x²)^(3/2) dx
= ∫ (a² + a² * tan²z)^(3/2) * (a * sec²z dz)
= ∫ (a² * sec²z)^(3/2) * (a * sec²z dz)
= ∫ (a³ * sec³z) * (a * sec²z dz)
= a⁴∫ sec⁵z dz = a⁴M
M = ∫ sec⁵z dz = ∫ sec³z * (sec²z dz) = ∫ sec³z d(tanz)
= sec³ztanz - ∫ tanz * 3sec²z * secztanz dz
= sec³ztanz - 3∫ sec³z(sec²z - 1) dz
= sec³ztanz - 3M + 3∫ sec³z dz
4M = sec³ztanz + 3N
M = (1/4)sec³ztanz + (3/4)N
N = ∫ sec³z dz = ∫ secz d(tanz)
= secztanz - ∫ tanz * secztanz dz
= secztanz - ∫ secz(sec²z - 1) dz
= secztanz - N + ∫ secz dz
2N = secztanz + ∫ secz dz
N = (1/2)secztanz + (1/2)∫ secz dz
∫ secz dz = ∫ secz * (secz + tanz)/(secz + tanz) dz = ∫ d(secz + tanz)/(secz + tanz)
= ln|secz + tanz| + C
M = (1/4)sec³ztanz + (3/4)N = (1/4)sec³ztanz + (3/4)[(1/2)secztanz + (1/2)∫ secz dz]
= (1/4)sec³ztanz + (3/8)secztanz + (3/8)ln|secz + tanz|
原式 = a⁴[(1/4)sec³ztanz + (3/8)secztanz + (3/8)ln|secz + tanz|] + C'
= (a⁴/4)[√(a² + x²)/a]³(x/a) + (3a⁴/8)[√(a² + x²)/a](x/a) + (3a⁴/8)ln|x/a + √(a² + x²)/a| + C'
= (x/4)(a² + x²)^(3/2) + (3a²x/8)√(a² + x²) + (3a⁴/8)ln|x + √(a² + x²)| + C
= (x/8)(5a² + 2x²)√(a² + x²) + (3a⁴/8)ln|x + √(a² + x²)| + C