Integration (Indefinite and Definite)

Integration as the Reverse of Differentiation

If $\frac{d}{dx}[F(x)] = f(x)$, then we say that $F(x)$ is an antiderivative (or integral) of $f(x)$. We write:

$\int f(x) \, dx = F(x) + c$

where $c$ is the constant of integration. This constant arises because the derivative of any constant is zero, so infinitely many functions share the same derivative.

For example, since $\frac{d}{dx}(x^3) = 3x^2$, we have $\int 3x^2 \, dx = x^3 + c$.

The Power Rule for Integration

The most fundamental integration rule at AS Level is:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + c \quad (n \neq -1)$

This works by reversing the power rule for differentiation. You increase the power by one and divide by the new power.

For a general term $ax^n$:

$\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + c$

Integration is linear, meaning you can integrate a sum or difference of terms by integrating each term separately:

$\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx$

Definite Integrals

A definite integral has upper and lower limits of integration and produces a numerical value:

$\int_a^b f(x) \, dx = \left[ F(x) \right]_a^b = F(b) - F(a)$

The constant of integration cancels when evaluating a definite integral, so we do not include $+c$.

The Fundamental Theorem of Calculus guarantees that this process correctly computes the signed area between the curve $y = f(x)$ and the $x$-axis, from $x = a$ to $x = b$.

Area Under a Curve

The definite integral $\int_a^b f(x) \, dx$ gives the signed area between the curve and the $x$-axis. When $f(x) \geq 0$ on $[a, b]$, the integral equals the area directly. When $f(x) < 0$ on part of the interval, the integral gives a negative contribution for that region.

To find the total area (always positive), you must:

1. Find where $f(x) = 0$ (the roots) within the interval.
2. Split the integral at these roots.
3. Take the absolute value of each part and add them together.

Preparing Expressions for Integration

Before integrating, you may need to rewrite expressions so that every term is in the form $ax^n$. Common manipulations include:

• Expanding brackets: $\int x(x + 3) \, dx = \int (x^2 + 3x) \, dx$
• Dividing terms: $\int \frac{x^3 + 2x}{x} \, dx = \int (x^2 + 2) \, dx$
• Rewriting roots: $\int \sqrt{x} \, dx = \int x^{1/2} \, dx$
• Rewriting reciprocals: $\int \frac{3}{x^2} \, dx = \int 3x^{-2} \, dx$

Finding the Constant of Integration

When given additional information (such as a point the curve passes through), you can determine the value of $c$. If $\int f(x) \, dx = F(x) + c$ and the curve passes through $(a, b)$, then $b = F(a) + c$, giving $c = b - F(a)$.

Tip 1: Always Include $+c$ for Indefinite Integrals

Forgetting the constant of integration is one of the most common mark losses. For indefinite integrals, always write $+c$ at the end. For definite integrals (with limits), you do not need $+c$.

Tip 2: Rewrite Before Integrating

If the integrand is not already a sum of terms like $ax^n$, rewrite it first. Expand products, divide fractions term by term, and convert roots and reciprocals to index notation before applying the power rule.

Tip 3: Use Square Brackets with Limits for Definite Integrals

When evaluating $\int_a^b f(x) \, dx$, write $\left[ F(x) \right]_a^b$ and then substitute the upper limit first, minus the lower limit. This notation scores method marks and keeps your working clear.

Tip 4: Check by Differentiating

After integrating, differentiate your answer. If you get back to the original integrand, your integration is correct. This is the most reliable self-check available.

Tip 5: Handle Negative Areas Carefully

When asked for an "area", remember that the definite integral gives signed area. If the curve dips below the $x$-axis, split the integral and take absolute values. A sketch of the curve helps you identify where $f(x) < 0$.