几道基础的积分题∫（X+1/(√x))(3√x)dx（X加上1除以根号X的和,乘以X的立方根）∫（5^（2x+2))/(5+5^(2x))dx(5的2X+2次方除以5加上5的2X次方的和）∫X/(√（x+3))dx(X除以根号下X+3的和）

Question

Answer 1

1、∫(x+1/(√x))(3√x)dx
=∫(x+x^(-1/2))x^(1/3)dx
=∫x·x^(1/3)dx+∫x^(-1/2)·x^(1/3)dx
=∫x^(4/3)dx+∫x^(-1/6)dx
=(3/7)x^(7/3)+(6/5)x^(5/6)+C
2、∫(5^(2x+2))/(5+5^(2x))dx
=25∫5^(2x)/(5+5^(2x))dx
=25(∫dx-5∫1/(5+5^(2x))dx)
令u=5+5^(2x),x=ln(u-5)/2ln5,dx=1/(2ln5(u-5))du
原式=25(x+(1/2ln5)∫-5/u(u-5)du)
=25(x+(1/2ln5)(∫1/udu-∫1/(u-5)du))
=25(x+ln(5+5^(2x))/2ln5-ln(5^(2x))/2ln5)+C
=12.5*log(5^(2x)+5)+C (log以5为底）
3、∫x/(√(x+3))dx
令u=√(x+3),x=u^2-3,dx=2udu
原式=∫(u^2-3)2u/udu
=2∫u^2du-6∫du
=2u^3/3-6u+C
=(2/3)√(x+3)^3-6√(x+3)+C