This method simplifies calculations by replacing the variable of integration with another variable.
Steps for Integration by Substitution
- Set u = g(x)
- Using du = g'(x)dx, express dx in terms of du
- Rewrite the integral in the form ∫f(g(x))g'(x)dx = ∫f(u)du
- After integrating, replace u with x
Example ①
Calculate ∫(2x+1)⁴ dx using substitution.
Answer:Let u = 2x+1. Then du = 2dx → dx = du/2。
∫u⁴ × (du/2) = (1/2) × u⁵/5 + C = (2x+1)⁵/10 + C
Calculate ∫(2x+1)⁴ dx using substitution.
Answer:Let u = 2x+1. Then du = 2dx → dx = du/2。
∫u⁴ × (du/2) = (1/2) × u⁵/5 + C = (2x+1)⁵/10 + C
Example ②
Calculate ∫x(x²+1)³ dx using substitution.
Answer:Let u = x²+1. Then du = 2xdx → xdx = du/2。
∫u³ × (du/2) = u⁴/8 + C = (x²+1)⁴/8 + C
Calculate ∫x(x²+1)³ dx using substitution.
Answer:Let u = x²+1. Then du = 2xdx → xdx = du/2。
∫u³ × (du/2) = u⁴/8 + C = (x²+1)⁴/8 + C
Key Points
- Set the “expression inside” equal to u, and calculate du
- Check that all dx terms can be converted into du
- The inverse operation of differentiating a composite function
Practice Problems
- Calculate ∫(3x−2)⁵ dx。
- Calculate ∫x²(x³+1)⁴ dx。
- Calculate ∫2x/√(x²+1) dx。
Answers and Explanations
- Answer:(3x−2)⁶/18 + C
Explanation: u=3x−2、du=3dx。∫u⁵du/3=u⁶/18+C。 - Answer:(x³+1)⁵/15 + C
Explanation: u=x³+1、du=3x²dx。∫u⁴du/3=u⁵/15+C。 - Answer:2√(x²+1) + C
Explanation: u=x²+1、du=2xdx。∫u^(−1/2)du=2u^(1/2)+C=2√(x²+1)+C。