Question

求积分：∫（r^3/(r^2+x^2)）dr

Answer 1

∫r³/(r²+x²)^(3/2) dr

令u=r²+x²

du=2r dr

原式=(1/2)∫(u-x²)/u^(3/2) du

=(1/2)∫[1/√u - x²/u^(3/2)] du

=(1/2)[2√u + 2x²/√u] + C

=√u + x²/√u + C

=(u+x²)/√u + C

=(r²+2x²)/√(r²+x²) + C