We cover examples involving two repetitions of integration by parts and recurrence forms that return to the same form.
Example ① (Twice) ∫ x²·e^x dx
Solution:= x²·e^x − ∫ 2x·e^x dx = x²·e^x − 2·{x·e^x − e^x} = (x²−2x+2)e^x + C
Solution:= x²·e^x − ∫ 2x·e^x dx = x²·e^x − 2·{x·e^x − e^x} = (x²−2x+2)e^x + C
Example ② (Recurrence) I = ∫ e^x·sin x dx
Solution:After integration by parts twice, I = e^x sin x − e^x cos x − I → 2I = e^x(sin x − cos x)
I = e^x(sin x − cos x)/2 + C
Solution:After integration by parts twice, I = e^x sin x − e^x cos x − I → 2I = e^x(sin x − cos x)
I = e^x(sin x − cos x)/2 + C
Key Point
- For e^x × trigonometric functions, two integrations by parts return to the original form → solve the equation
Practice Problems
- ∫ x·log x dx
Answer and Explanation
- Answer:(x²/2)log x − x²/4 + C
f=log x、g'=x.