In calculus, an integral represents the area under a curve or the accumulation of quantities. There are two main types: indefinite integrals (antiderivatives), which find the original function from its derivative, and definite integrals, which calculate the net area between a function and the x-axis over a specific interval [a,b]. The Fundamental Theorem of Calculus connects differentiation and integration.

To solve an indefinite integral ∫f(x)dx, you need to find a function F(x) whose derivative is f(x). This is called the antiderivative. Common techniques include: power rule (∫x^n dx = x^(n+1)/(n+1) + C for n≠-1), substitution method, integration by parts (∫u dv = uv – ∫v du), partial fractions for rational functions, and trigonometric identities. Always add the constant of integration ‘+C’ since derivatives eliminate constants.

An indefinite integral ∫f(x)dx represents a family of functions (antiderivatives) and includes a constant ‘+C’. A definite integral ∫[a to b] f(x)dx represents a specific number—the net area under the curve from x=a to x=b—and does not include ‘+C’. The definite integral is evaluated using the Fundamental Theorem: ∫[a to b] f(x)dx = F(b) – F(a), where F is any antiderivative of f.

Integration by parts is based on the product rule for derivatives and uses the formula: ∫u dv = uv – ∫v du. To apply it: 1) Choose u and dv from your integrand, 2) Compute du (derivative of u) and v (integral of dv), 3) Substitute into the formula. Good choices for ‘u’ follow the LIATE rule: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential functions—in that order of preference.

No, not all functions have elementary antiderivatives that can be expressed using standard mathematical functions. Examples include ∫e^(-x²)dx (error function), ∫sin(x)/x dx (sine integral), and ∫√(1-x⁴)dx (elliptic integral). These require special functions or numerical methods for evaluation. However, most functions encountered in basic calculus courses can be integrated using standard techniques.

Common integration techniques include: 1) Basic rules (power rule, linearity), 2) Substitution (u-substitution) for composite functions, 3) Integration by parts for products of functions, 4) Partial fraction decomposition for rational functions, 5) Trigonometric identities and substitutions for trigonometric integrals, 6) Completing the square for quadratic denominators, and 7) Special formulas for standard forms like ∫1/(x²+a²)dx = (1/a)arctan(x/a) + C.

To verify your integral solution, take the derivative of your answer and see if you get back the original integrand. For example, if you found ∫2x dx = x² + C, then d/dx(x² + C) = 2x, which matches the original function. For definite integrals, you can also approximate the area numerically (using rectangles, trapezoids, etc.) and compare with your exact result.

We add ‘+C’ (the constant of integration) to indefinite integrals because the derivative of any constant is zero. This means that if F(x) is an antiderivative of f(x), then F(x) + C is also an antiderivative for any constant C. Since we don’t know which specific antiderivative was the original function, we include ‘+C’ to represent the entire family of possible antiderivatives.