Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) states that integration is the inverse process of differentiation. If a function $f(x)$ is the derivative of $F(x)$, then the integral of $f(x)$ 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 evaluates the net change of the antiderivative over a specific interval $[a, b]$, expressed as $\int_a^b f'(x) dx = f(b) - f(a)$.

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

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

In definite integration, this constant is not required because it cancels out during the subtraction of the upper and lower limits: $(F(b) + c) - (F(a) + c) = F(b) - F(a)$.

Step 1: Preparation: Rewrite the integrand into an 'integrable form' by expanding brackets or converting roots and fractions into powers of $x$ (e.g., $\frac{1}{x^2}$ becomes $x^{-2}$).

Step 2: Integration: Apply the power rule by increasing the exponent by 1 and dividing by the new exponent: $\int x^n dx = \frac{x^{n+1}}{n+1} + c$ (for $n \neq -1$).

Step 3: Evaluation (Definite Only): Place the integrated expression in square brackets $[...]_a^b$, then substitute the upper limit $b$ and subtract the result of substituting the lower limit $a$.

Step 4: Finding 'c': If given a specific coordinate $(x, y)$ that the curve passes through, substitute these values into the indefinite integral to solve for the specific value of $c$.

Check the Integrand: Always ensure the function is fully expanded and in power form before integrating. A common mistake is trying to integrate terms inside brackets without expanding them first.

Bracket Management: When evaluating definite integrals, always put the substitution of the lower limit in its own set of brackets to avoid sign errors during subtraction.

Calculator Verification: Use the integration function on a scientific calculator to check your numerical answers for definite integrals, but remember that exams usually require you to show the algebraic steps.

Sketching: If asked for an area, always sketch the curve first to identify if any part of the region falls below the x-axis, which would result in a negative integral value.

Negative Areas: A definite integral calculates the 'signed area'. If a region is below the x-axis, the integral will be negative. To find the physical area, you must take the absolute value (modulus).

Crossing the Axis: If an area consists of sections both above and below the x-axis, integrating across the whole interval will cause the positive and negative areas to cancel each other out. You must integrate the sections separately.

The $n = -1$ Case: The standard power rule does not apply to $x^{-1}$ because it would result in division by zero. This requires logarithmic integration, which is a separate concept.