14-December-2021 Homework

    Find the integrals. (You can use any method you learned from the course. Not all problems require integration by parts.)

    1. hw141221 - 图1
    2. hw141221 - 图2%20%5C%2C%20dx#card=math&code=%5Cint%20%5Cln%282x%2B1%29%20%5C%2C%20dx&id=Ih7Gp)
    3. hw141221 - 图3%20%5C%2C%20dx%20%3D%20%5Cint%20%5Ctan%5E%7B-1%7D(x)%5C%2C%20dx#card=math&code=%5Cint%20%5Carctan%28x%29%20%5C%2C%20dx%20%3D%20%5Cint%20%5Ctan%5E%7B-1%7D%28x%29%5C%2C%20dx&id=T1vIT)
    4. hw141221 - 图4%5C%2C%20dx#card=math&code=%5Cint%20x%20%5Csin%282x%29%5C%2C%20dx&id=V9LDV)
    5. hw141221 - 图5%5C%2C%20dx#card=math&code=%5Cint%20x%20%5Ccos%282x%29%5C%2C%20dx&id=xbJQF)
    6. hw141221 - 图6
    7. hw141221 - 图7
    8. hw141221 - 图8%20%5C%2C%20dx%2C%20%5Cquad%20I_2%20%3D%20%5Cint%20%5Ccos(%5Cln%20x)%5C%2C%20dx#card=math&code=I_1%20%3D%20%5Cint%20%5Csin%20%28%5Cln%20x%29%20%5C%2C%20dx%2C%20%5Cquad%20I_2%20%3D%20%5Cint%20%5Ccos%28%5Cln%20x%29%5C%2C%20dx&id=jR3Ii) (Hint: let hw141221 - 图9, or you can use integration by parts and simultaneously solve out these two integrals)
    9. hw141221 - 图10%5E2%5C%2C%20dx#card=math&code=%5Cint%20%28%5Cln%20x%29%5E2%5C%2C%20dx&id=S6cHx)
    10. hw141221 - 图11%20%5C%2C%20dx%20%3D%20%5Cint%20%5Csin%5E%7B-1%7D%20x%20%5C%2C%20dx#card=math&code=%5Cint%20%5Carcsin%28x%29%20%5C%2C%20dx%20%3D%20%5Cint%20%5Csin%5E%7B-1%7D%20x%20%5C%2C%20dx&id=u9U0O)
    11. hw141221 - 图12%5C%2C%20dx%20%3D%20%5Cint%20x%20%5Ctan%5E%7B-1%7D%20(x)%20%5C%2C%20dx#card=math&code=%5Cint%20x%20%5Carctan%28x%29%5C%2C%20dx%20%3D%20%5Cint%20x%20%5Ctan%5E%7B-1%7D%20%28x%29%20%5C%2C%20dx&id=a11Jk)
    12. hw141221 - 图13

    Derive recursive formulas for the following integrals using the technique of integration by parts. Assume that hw141221 - 图14 is a positive integer.

    1. hw141221 - 图15
    2. hw141221 - 图16%5C%2C%20dx#card=math&code=%5Cint%20x%5En%20%5Ccos%20%28x%29%5C%2C%20dx&id=X8YeD)
    3. hw141221 - 图17%5C%2C%20dx#card=math&code=%5Cint%20%5Ctan%5En%28x%29%5C%2C%20dx&id=nDjbC) (Hint: hw141221 - 图18, and hence hw141221 - 图19.)

    Please write down the numbering.

    Email your homework to: yqfeng@stu.edu.cn with subject “hw141221”. You can submit a picture or a scanned copy. Due date: Thursday 23-December, 23:59 (GMT+8).