Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that integration is the inverse process of differentiation. If a function $f(x)$ is the derivative of $F(x)$, then integrating $f(x)$ returns the original function $F(x)$ plus a constant.

Indefinite Integration refers to finding the general form of the antiderivative, expressed as $\int f(x) \, dx = F(x) + c$.

Definite Integration calculates the net change or area between two specific boundaries, known as the limits of integration, expressed as $\int_a^b f(x) \, dx$.

When a function is differentiated, any constant terms disappear because the gradient of a horizontal line ($y = c$) is always zero. For example, $y = 2x + 5$ and $y = 2x - 10$ both have the same derivative, $\frac{dy}{dx} = 2$.

Because information is lost during differentiation, we must add a constant of integration ($+c$) when performing indefinite integration to represent all possible original functions.

This constant $c$ represents the vertical shift of the function on a coordinate plane, which does not affect the slope at any given point.

Check for +c: In any question asking for an 'integral' or 'antiderivative' without limits, you will lose marks if you forget the constant of integration.

Order of Subtraction: Always perform $F(\text{upper}) - F(\text{lower})$. Reversing this order will result in the correct magnitude but the wrong sign.

Integrand Simplification: Before integrating, always rewrite terms into a power form ($x^n$). For example, convert $\frac{1}{x^2}$ to $x^{-2}$ and $\sqrt{x}$ to $x^{1/2}$ to apply standard rules easily.

Calculator Verification: Use the definite integral function on a scientific calculator to verify your numerical answers, but ensure you show the manual substitution steps to gain full method marks.

The 'Vanishing' Constant: Students often forget that constants differentiate to zero, leading them to believe they can determine the exact original function without additional coordinate information.

Bracket Errors: When subtracting $F(a)$, students frequently fail to distribute the negative sign across all terms of the lower limit evaluation, leading to arithmetic errors.

Variable Confusion: The $dx$ notation is not just a decoration; it identifies the variable of integration. Any other letters in the integrand must be treated as constants.