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)
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/27 02:19:04
![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)](/uploads/image/z/9989634-66-4.jpg?t=%E5%87%A0%E9%81%93%E5%9F%BA%E7%A1%80%E7%9A%84%E7%A7%AF%E5%88%86%E9%A2%98%E2%88%AB%EF%BC%88X%2B1%2F%28%E2%88%9Ax%29%29%283%E2%88%9Ax%29+dx+%EF%BC%88X%E5%8A%A0%E4%B8%8A1%E9%99%A4%E4%BB%A5%E6%A0%B9%E5%8F%B7X%E7%9A%84%E5%92%8C%2C%E4%B9%98%E4%BB%A5X%E7%9A%84%E7%AB%8B%E6%96%B9%E6%A0%B9%EF%BC%89%E2%88%AB%EF%BC%885%5E%EF%BC%882x%2B2%29%29%2F%285%2B5%5E%282x%29%29+dx+%285%E7%9A%842X%2B2%E6%AC%A1%E6%96%B9%E9%99%A4%E4%BB%A55%E5%8A%A0%E4%B8%8A5%E7%9A%842X%E6%AC%A1%E6%96%B9%E7%9A%84%E5%92%8C%EF%BC%89%E2%88%ABX%2F%28%E2%88%9A%EF%BC%88x%2B3%29%29+dx+%28X%E9%99%A4%E4%BB%A5%E6%A0%B9%E5%8F%B7%E4%B8%8BX%2B3%E7%9A%84%E5%92%8C%EF%BC%89)
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)
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)
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)
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