Integration

Integration answers two closely related questions:

1. Which function has this derivative? This gives an indefinite integral, or family of antiderivatives.
2. How much has accumulated over an interval? This gives a definite integral, used for signed area, displacement, total change and probability.

The fundamental theorem of calculus connects these questions. If $F'(x)=f(x)$, then

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

This section develops the complete A-level pathway from reversing differentiation to solving separable differential equations.

Prerequisites

Before starting, you should be able to:

• simplify powers and algebraic fractions
• expand, factorise and complete the square
• use trigonometric identities
• manipulate exponential and logarithmic expressions
• differentiate powers, trigonometric functions, exponentials and logarithms
• apply the product and chain rules

If differentiation is uncertain, begin with A-level calculus and differentiation. For algebraic gaps, revisit Algebra and functions and Exponentials and logarithms.

The central idea

Since

$$\frac{d}{dx}(x^3)=3x^2,$$

reversing the process gives

$$\int 3x^2\,dx=x^3+C.$$

The constant $C$ is essential because every function $x^3+C$ has derivative $3x^2$. Differentiation loses constants, so indefinite integration must restore the whole family.

For $n\ne-1$, reversing the power rule gives

$$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C.$$

The exception $n=-1$ matters:

$$\int \frac1x\,dx=\ln|x|+C,$$

not $x^0/0$. The absolute value allows the antiderivative on intervals where $x<0$ as well as where $x>0$.

Worked example: indefinite and definite integration

Find $\int(6x^2-4x+3)\,dx$, then evaluate $\int_1^2(6x^2-4x+3)\,dx$.

Integrate term by term:

$$\begin{aligned} \int(6x^2-4x+3)\,dx &=6\left(\frac{x^3}{3}\right)-4\left(\frac{x^2}{2}\right)+3x+C\\ &=2x^3-2x^2+3x+C. \end{aligned}$$

Let $F(x)=2x^3-2x^2+3x$. For the definite integral, the arbitrary constant is unnecessary because it cancels:

$$\begin{aligned} \int_1^2(6x^2-4x+3)\,dx &=[2x^3-2x^2+3x]_1^2\\ &=(16-8+6)-(2-2+3)\\ &=11. \end{aligned}$$

Check the indefinite answer by differentiating it. Check the definite answer roughly: the integrand is positive and lies between $5$ and $19$ on an interval of width $1$, so $11$ is plausible.

Integral, area and total change

The value $\int_a^b f(x)\,dx$ is a signed accumulation. Regions above the $x$-axis contribute positively and regions below it contribute negatively. Therefore

$$\int_a^b f(x)\,dx$$

is not automatically the geometric area between the curve and the axis.

For example,

$$\int_{-1}^{1}x\,dx=0,$$

because the negative and positive triangular regions cancel. Their total geometric area is

$$\int_{-1}^{0}(-x)\,dx+\int_0^1x\,dx=1.$$

The same distinction appears in mechanics: integrating velocity gives displacement, whereas integrating speed gives distance travelled. Study definite integrals and areas before tackling questions in which a curve crosses an axis or two curves intersect.

Recommended learning pathway

Follow this order if you are learning integration for the first time.

1. Integration basics introduces antiderivatives, the constant of integration and standard integrals.
2. Integration as a limit explains accumulation using sums of increasingly narrow strips and connects the notation $\int f(x)\,dx$ to a limiting process.
3. Definite integrals and areas covers the fundamental theorem, signed area, total area and areas between curves.
4. The trapezium rule approximates a definite integral when an antiderivative is unavailable or values are given in a table.
5. Integration by substitution reverses the chain rule and handles composite expressions.
6. Integration by parts reverses the product rule and is especially useful for products involving polynomials, exponentials, logarithms and trigonometric functions.
7. Integration using partial fractions decomposes rational functions into standard integrable forms.
8. Separable differential equations uses integration to recover a function from a relationship between its rate of change and its variables.
9. Interpreting differential equation models develops initial conditions, constants, parameters, solution families and the limitations of models.

Diagnostic readiness check

Try these without notes. Each answer identifies the lesson to study next.

1. Reverse a derivative

Find $\int(4x^3-3x^{-2})\,dx$.

Answer

$$\int(4x^3-3x^{-2})\,dx=x^4+3x^{-1}+C=x^4+\frac3x+C.$$

For the second term, increase the power from $-2$ to $-1$, then divide by $-1$. The two negative signs cancel. Differentiate the answer to check it. If this was difficult, start with integration basics.

2. Use an initial condition

Given $\frac{dy}{dx}=6x-4$ and $y=5$ when $x=1$, find $y$ in terms of $x$.

Answer

First integrate:

$$y=3x^2-4x+C.$$

Now use the complete family, not an expression with $C$ omitted:

$$5=3(1)^2-4(1)+C,$$

so $C=6$. Therefore

$$\boxed{y=3x^2-4x+6}.$$

3. Interpret a definite integral

A particle has velocity $v(t)=3t^2-12t+9$ metres per second. What does $\int_0^4v(t)\,dt$ represent?

Answer

It represents displacement from $t=0$ to $t=4$, not necessarily distance travelled.

$$\begin{aligned} \int_0^4(3t^2-12t+9)\,dt &=[t^3-6t^2+9t]_0^4\\ &=64-96+36\\ &=4\text{ m}. \end{aligned}$$

Since $v(t)=3(t-1)(t-3)$ changes sign at $t=1$ and $t=3$, finding distance would require splitting the interval there and making each contribution positive.

4. Choose a method

Match each integral to its most natural main method.

$$\text{(a) }\int 2x\cos(x^2)\,dx, \qquad \text{(b) }\int xe^x\,dx, \qquad \text{(c) }\int\frac{1}{x(x+1)}\,dx.$$

Answer

• (a) substitution, because $2x$ is the derivative of the inner function $x^2$
• (b) integration by parts, because the integrand is a product and differentiating $x$ simplifies it
• (c) partial fractions, because the denominator is a factorised polynomial

Method recognition is a major A-level skill. A correct integration method often begins by recognising which differentiation rule or algebraic form is being reversed.

Common misconceptions

• Forgetting $C$. Include $+C$ in every indefinite integral. Do not attach it to a definite integral.
• Using the power rule when $n=-1$. The integral of $x^{-1}$ is $\ln|x|+C$.
• Treating signed area as total area. Split at every crossing and make below-axis contributions positive when geometric area is requested.
• Substituting the lower limit first and reversing the subtraction. The convention is always $F(b)-F(a)$.
• Assuming every product needs integration by parts. A product such as $2x\cos(x^2)$ is structured for substitution.
• Giving only a decimal when an exact value is available. Preserve logarithms, surds, $\pi$ and rational values unless the question requests an approximation.

How to check an integral

Differentiate an indefinite answer. The result should reproduce the original integrand exactly. For a definite integral, also inspect signs, interval width and approximate function values. In an area question, sketch the curves and mark all intersections before integrating. In a differential equation, substitute the final function back into the original equation and verify any initial condition.

Strong students should be able to move between four views of integration: reverse differentiation, signed area, accumulated change and the limiting sum of small contributions. Tutors can use the diagnostic questions to distinguish a conceptual difficulty from an algebra error or a failure to recognise the required method.