置換積分 不定積分

質問

(1) ∫(1+logx)/x dx (t=logx)

(2) ∫x³√(1+x²) dx (t=1+x²)

解答お願いします途中式もお願いします

回答No.1

info222_ · ベストアンサー

(1)
I=∫(1+log(x))/x dx (t=log(x), dt=(1/x)・dx)
=∫(1+t) dt
=t+t^2/2+C
=log(x)+(1/2)(log(x))^2+C(C:任意定数)

(2)
I= ∫x³√(1+x²) dx (t=1+x², dt=2xdx)
=∫ x^2・(1+x^2)^(1/2)・xdx
=∫ (t-1)・t^(1/2)・dt/2
=(1/2)∫ (t^(3/2)-t^(1/2))dt
=(1/2)[(2/5)t^(5/2)-(2/3)t^(3/2)]+C
=(1/5)t^2・t^(1/2)-(1/3)t・t^(1/2)+C
={(1/5)(1+x^2)^2-(1/3)(1+x^2)}√(1+x^2) +C (C:任意定数)
or
=(1/15)(3x^4+x^2-2)√(1+x^2) +C (C:任意定数)
or
=(1/15)(3x^2-2)(x^2+1)√(1+x^2)+C (C:任意定数)