These are comprehensive exercises on indefinite integrals, integration by substitution, and integration by parts.
Example ①
Calculate ∫(x+1)√x dx.
Answer:Product with x^(1/2)。∫(x^(3/2)+x^(1/2))dx = (2/5)x^(5/2) + (2/3)x^(3/2) + C = (2/5)x²√x + (2/3)x√x + C
Calculate ∫(x+1)√x dx.
Answer:Product with x^(1/2)。∫(x^(3/2)+x^(1/2))dx = (2/5)x^(5/2) + (2/3)x^(3/2) + C = (2/5)x²√x + (2/3)x√x + C
Example ②
State the method for ∫(2x−1)/(x²−x+1)² dx.
Answer:Let u = x²−x+1. Then du = (2x−1)dx。
∫u⁻² du = −u⁻¹ + C = −1/(x²−x+1) + C
State the method for ∫(2x−1)/(x²−x+1)² dx.
Answer:Let u = x²−x+1. Then du = (2x−1)dx。
∫u⁻² du = −u⁻¹ + C = −1/(x²−x+1) + C
Summary
- Basic: ∫xⁿdx = xⁿ⁺¹/(n+1)+C
- Substitution: Convert using u=g(x) and du=g'(x)dx
- Integration by parts: ∫uv' = uv−∫u'v
- Determine the constant of integration from the initial values
Practice Problems
- Calculate ∫x/√(1+x²) dx.
- Calculate ∫(x²+2x)(x+1) dx by integration by parts or expansion.
- Find f(x) satisfying f'(x) = (2x+1)² and f(0) = 1.
Answers and Explanations
- Answer:√(1+x²) + C
Explanation: u=1+x²、du=2xdx。∫(1/2)u^(−1/2)du=u^(1/2)+C=√(1+x²)+C。 - Answer:x⁴/4+x³+x²+C (Expansion: (x²+2x)(x+1)=x³+3x²+2x。Integration: x⁴/4+x³+x²+C)
Explanation: Expand first, then integrate. - Answer:f(x) = (2x+1)³/6 + 5/6
Explanation: ∫(2x+1)²dx=(2x+1)³/6+C。f(0)=1/6+C=1→C=5/6。