Substitution integration: a method of changing variables by setting u = g(x) and using du = g'(x)dx。
∫ f(g(x))·g'(x) dx = ∫ f(u) du
Example ① ∫ (2x+1)⁵ dx
Solution:u = 2x+1、du = 2dx → dx = du/2。
∫ u⁵·(du/2) = u⁶/12 + C = (2x+1)⁶/12 + C
Solution:u = 2x+1、du = 2dx → dx = du/2。
∫ u⁵·(du/2) = u⁶/12 + C = (2x+1)⁶/12 + C
Example ② ∫ x·√(x²+1) dx
Solution:u = x²+1、du = 2x dx → x dx = du/2。
∫ √u·(du/2) = (1/3)u^{3/2} + C = (x²+1)√(x²+1)/3 + C
Solution:u = x²+1、du = 2x dx → x dx = du/2。
∫ √u·(du/2) = (1/3)u^{3/2} + C = (x²+1)√(x²+1)/3 + C
Key Point
- The basic idea is to “set the inside expression equal to u”
- Choose u so that the form du appears
Practice Problems
- ∫ cos(3x) dx
- ∫ x·e^{x²} dx
Solutions and Explanations
- Solution:(1/3)sin(3x) + C
- Solution:(1/2)e^{x²} + C