Derived from the product rule (f·g)' = f'·g + f·g'
∫ f·g' dx = f·g − ∫ f'·g dx is called integration by parts.
Guideline for choosing: logarithms and powers of x should be placed as f (the factor to differentiate), while trigonometric functions and e^x should be placed as g' (the factor to integrate); this often makes the calculation easier.
Example ① ∫ x·e^x dx
Solution:f=x、g'=e^x → f'=1、g=e^x.
∫ x·e^x dx = x·e^x − ∫ 1·e^x dx = x·e^x − e^x + C = (x−1)e^x + C
Solution:f=x、g'=e^x → f'=1、g=e^x.
∫ x·e^x dx = x·e^x − ∫ 1·e^x dx = x·e^x − e^x + C = (x−1)e^x + C
Example ② ∫ log x dx
Solution:f=log x、g'=1 → f'=1/x、g=x.
= x log x − ∫ x·(1/x) dx = x log x − x + C
Solution:f=log x、g'=1 → f'=1/x、g=x.
= x log x − ∫ x·(1/x) dx = x log x − x + C
Key Point
- “Choose the factor whose derivative becomes simpler as f” and “the factor whose integral is simpler as g'”
- Place log in f, x^n in f, and e^x or sin/cos in g'
Practice Problems
- ∫ x·sin x dx
Answer and Explanation
- Answer:−x cos x + sin x + C
f=x、g'=sin x → g=−cos x.