HSC Maths Integration: Complete Guide with Worked Examples

Last updated: 20 July 2026

Integration is worth 15-20% of the HSC Maths Advanced exam, and it's where a lot of students lose marks they shouldn't. The concept isn't hard, but the details (signs, limits, + C) trip students up constantly.

This guide covers everything you need: the basic rules, area problems, the trapezoidal rule, and (for Extension 1) integration by substitution.

What is integration?

Integration is the reverse of differentiation. If differentiating gives you the rate of change, integrating gives you the accumulated total.

There are two types:

• Indefinite integrals (no limits): give a general formula with + C
• Definite integrals (with limits): give a numerical value (the area under the curve between two points)

Basic integration rules

These are the rules you need to know cold. Some are on the formula sheet, but you should recognise them instantly.

Power rule

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

Add 1 to the power, divide by the new power, add + C.

Example: ∫5x³ dx = 5x⁴/4 + C = 5x⁴/4 + C

Example: ∫1/√x dx = ∫x^(-1/2) dx = x^(1/2)/(1/2) + C = 2√x + C

Reciprocal

$$\int \frac{1}{x} \, dx = \ln|x| + C$$

This is the exception to the power rule (n = -1 doesn't work). Note the absolute value around x.

Exponential

$$\int e^x \, dx = e^x + C$$

The exponential function is its own integral (and its own derivative). Easy to remember.

Trigonometric

$$\int \sin x \, dx = -\cos x + C$$ $$\int \cos x \, dx = \sin x + C$$

Watch the negative sign on the sine integral. This is the most common integration error in HSC exams.

Definite integrals

A definite integral calculates the exact area under a curve between two limits.

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

where F(x) is the antiderivative of f(x).

Example: Calculate ∫₀² (3x² + 2x) dx

Antiderivative: F(x) = x³ + x² + C F(2) = 8 + 4 = 12 F(0) = 0 + 0 = 0 Result: 12 - 0 = 12

With definite integrals, the + C cancels out (you subtract one C from another), so you don't need to write it.

Area under a curve

The area between the curve y = f(x), the x-axis, and the lines x = a and x = b is:

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

This only works when f(x) is above the x-axis. If f(x) is below the x-axis, the integral gives a negative value. The actual area is the absolute value.

Area between two curves

The area between y = f(x) and y = g(x) from x = a to x = b is:

$$A = \int_a^b [f(x) - g(x)] \, dx$$

where f(x) is the upper curve and g(x) is the lower curve.

Common mistake: Getting the order wrong. If you subtract the upper curve from the lower curve, you get a negative area. Always check which curve is on top.

Fix: Sketch the graphs. Label which is upper and which is lower. If they cross between the limits, split the integral at the intersection point.

The trapezoidal rule

When you can't integrate a function exactly (or the function is given as data points), use the trapezoidal rule to approximate the area:

$$A \approx \frac{h}{2}[f(a) + f(b)]$$

For multiple subintervals (n strips of width h):

$$A \approx \frac{h}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$$

where h = (b - a)/n and x₀ = a, x₁ = a + h, ..., xₙ = b.

Key points:

• The first and last function values have a coefficient of 1
• All middle values have a coefficient of 2
• h is the width of each strip

Common mistake: Using h instead of h/2, or getting the coefficients wrong.

Integration by substitution (Extension 1)

This is the main new integration technique in Extension 1. You substitute a new variable to simplify the integral.

The method

1. Choose a substitution u = g(x) (usually the "inside" function)
2. Calculate du/dx, then rearrange to get dx = du / g'(x)
3. Substitute into the integral, replacing all x terms with u terms
4. Integrate with respect to u
5. Substitute back to get the answer in terms of x

Example: Calculate ∫2x(x² + 1)⁴ dx

Let u = x² + 1 du/dx = 2x, so dx = du/(2x) Substitute: ∫2x · u⁴ · du/(2x) = ∫u⁴ du = u⁵/5 + C Substitute back: (x² + 1)⁵/5 + C

The 2x terms cancel perfectly. This is what makes substitution work: if you choose the right substitution, the extra terms cancel.

When to use substitution

Look for an integral where:

• You have a function and its derivative (or a multiple of its derivative) present
• There's a composite function (something raised to a power, or inside a trig function)

The substitution is usually the "inside" function.

Common substitution types

• u = (linear expression), e.g., u = 2x + 3
• u = (expression raised to a power), e.g., u = x² + 1
• u = sin(x) or u = cos(x) in trig integrals
• u = eˣ in exponential integrals

Common mistakes

Forgetting + C

Every indefinite integral needs + C. Every time. No exceptions.

Fix: Write + C as soon as you write the integral sign. Don't wait until the end.

Sign error with sine

∫sin(x) dx = -cos(x) + C. The negative sign. Don't forget it.

Fix: Test it mentally. The derivative of -cos(x) is sin(x). Correct. The derivative of cos(x) is -sin(x). Wrong sign.

Wrong limits after substitution

When you substitute u = g(x), you must also change the limits. If the original limits are x = a and x = b, the new limits are u = g(a) and u = g(b).

Fix: Write the new limits explicitly. Don't try to convert back to x in the middle of the calculation.

Area below the x-axis

If part of the curve is below the x-axis, the integral gives a negative value for that section. The area is the absolute value, but students often just report the integral value.

Fix: Check whether the function crosses the x-axis between the limits. If it does, split the integral and take absolute values.

Not checking the answer

After integrating, you can always check by differentiating your answer. If you get back to the original function, you're correct.

Fix: For the last question on every practice set, differentiate your integral answer to verify. Build the habit.

How to practice

1. Start with basic power rule integrals until automatic
2. Add trig and exponential integrals
3. Practice definite integrals (finding areas)
4. Do area-between-curves questions (sketch every time)
5. Practice the trapezoidal rule (check your arithmetic)
6. For Extension 1: do 20+ substitution problems until the pattern is automatic
7. Finish with HSC exam questions

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Based on the NESA HSC Mathematics Advanced and Extension 1 syllabuses. PracticePapers.io is not affiliated with NESA.