Master fundamental integration techniques including indefinite and definite integrals, substitution methods, integration by parts, and geometric interpretations. Learn to apply these techniques to solve complex calculus problems and real-world applications.
Use u-substitution to find the integral!
Find the integral:
Use integration by parts to find the integral!
Find the integral:
An antiderivative F of a function f satisfies F'(x) = f(x). The indefinite integral ∫f(x) dx represents the family of all antiderivatives, which differ by a constant C.
Find: ∫x³ dx
Step 1: Apply power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Step 2: n = 3, so n+1 = 4
Answer: x⁴/4 + C
Find: ∫5x² dx
Step 1: Factor out constant: 5∫x² dx
Step 2: Apply power rule: 5(x³/3) + C
Answer: 5x³/3 + C
Find: ∫(3x² + 2x + 1) dx
Step 1: Integrate each term separately
Step 2: ∫3x² dx + ∫2x dx + ∫1 dx = x³ + x² + x + C
Answer: x³ + x² + x + C