For definite integrals, when variables are substituted, the limits of integration must also be changed.
Example ① (Substitution) ∫_0^1 x·e^{x²} dx
Solution:u = x²、du = 2x dx → x dx = du/2. Limits: x=0→u=0、x=1→u=1.
= ∫_0^1 (1/2)e^u du = (1/2)[e^u]_0^1 = (e−1)/2
Solution:u = x²、du = 2x dx → x dx = du/2. Limits: x=0→u=0、x=1→u=1.
= ∫_0^1 (1/2)e^u du = (1/2)[e^u]_0^1 = (e−1)/2
Example ② (Integration by Parts) ∫_0^1 x·e^x dx
Solution:[x·e^x]_0^1 − ∫_0^1 e^x dx = e − (e−1) = 1
Solution:[x·e^x]_0^1 − ∫_0^1 e^x dx = e − (e−1) = 1
Key Point
- After substitution, be sure to change the limits of integration as well
- Integration by parts is [f·g]_a^b − ∫_a^b f'·g dx
Practice Problems
- ∫_0^{π/2} sin x · cos x dx
Answer and Explanation
- Answer:1/2
u=sin x、du=cos x dx. ∫_0^1 u du = 1/2.