Standard Integration Rules

Beyond the basic power rule, you need to know several standard integrals for the HSC. These are the building blocks for more complex integration.

Function f(x)	Integral ∫ f(x) dx
xⁿ (n ≠ −1)	xⁿ⁺¹/(n+1) + C
1/x	ln\|x\| + C
e^x	e^x + C
e^ax	(1/a)e^ax + C
sin(x)	−cos(x) + C
cos(x)	sin(x) + C
sec²(x)	tan(x) + C

Properties of Integrals

These properties allow you to break down complex integrals into simpler parts:

Constant Multiple Rule

\int k \cdot f(x) dx = k \cdot \int f(x) dx

Constants can be factored out of the integral.

Sum/Difference Rule

\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx

Integrate term by term.

Example: ∫ (5e^x − 3sin(x) + 2x) dx

= 5 · ∫ e^x dx − 3 · ∫ sin(x) dx + 2 · ∫ x dx

= 5e^x − 3(−cos(x)) + 2(x²/2) + C

= 5e^x + 3cos(x) + x² + C

Finding Particular Solutions

When given an initial condition (a known point on the curve), you can find the exact value of C and determine a particular solution rather than a general one.

Example: f'(x) = 3x² and f(1) = 5. Find f(x).

Step 1: Integrate: f(x) = ∫ 3x² dx = x³ + C

Step 2: Use the initial condition: f(1) = 1³ + C = 5, so C = 4.

Answer: f(x) = x³ + 4

Key Vocabulary

Indefinite Integral

An integral without limits, giving a family of functions differing by a constant C.

Primitive Function

Another name for an anti-derivative: a function whose derivative equals the integrand.

Initial Condition

A known value of the function at a specific point, used to determine C.

General Solution

The full family of anti-derivatives including +C, before applying initial conditions.

Worked Examples

1

Find ∫ (4x³ − 6x + 1/x) dx.

Step 1: Integrate each term: ∫ 4x³ dx = x⁴

Step 2: ∫ −6x dx = −3x²

Step 3: ∫ 1/x dx = ln|x|

Answer: x⁴ − 3x² + ln|x| + C

2

Find ∫ (e^2x + cos(x)) dx.

Step 1: ∫ e^2x dx = (1/2)e^2x (using the rule for e^ax)

Step 2: ∫ cos(x) dx = sin(x)

Answer: (1/2)e^2x + sin(x) + C

3

Given f'(x) = 2x − 1 and f(2) = 3, find f(x).

Step 1: Integrate: f(x) = x² − x + C

Step 2: Apply initial condition: f(2) = 4 − 2 + C = 3, so C = 1.

Answer: f(x) = x² − x + 1

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

What is ∫ e^3x dx?

3e^3x + C (1/3)e^3x + C e^3x + C e^3x/x + C

Question 2

What is ∫ sin(x) dx?

cos(x) + C −sin(x) + C −cos(x) + C sin(x) + C

Question 3

What is ∫ 1/x dx?

x⁰ + C −1/x² + C ln(x²) + C ln|x| + C

Question 4

If f'(x) = 6x and f(0) = 4, what is f(x)?

3x² + 4 6x² + 4 3x² 3x² − 4

Question 5

What is ∫ (3cos(x) + 2/x) dx?

−3sin(x) + 2ln|x| + C 3sin(x) + 2ln|x| + C 3sin(x) + 2/x² + C 3sin(x) − 2ln|x| + C

Indefinite Integrals

Standard Integration Rules

Properties of Integrals

Example: ∫ (5e^x − 3sin(x) + 2x) dx

Finding Particular Solutions

Example: f'(x) = 3x² and f(1) = 5. Find f(x).

Key Vocabulary

Indefinite Integral

Primitive Function

Initial Condition

General Solution

Worked Examples

Find ∫ (4x³ − 6x + 1/x) dx.

Find ∫ (e^2x + cos(x)) dx.

Given f'(x) = 2x − 1 and f(2) = 3, find f(x).

Knowledge Check

Key Concepts Summary

Indefinite Integrals

Standard Integration Rules

Properties of Integrals

Example: ∫ (5ex − 3sin(x) + 2x) dx

Finding Particular Solutions

Example: f'(x) = 3x2 and f(1) = 5. Find f(x).

Key Vocabulary

Indefinite Integral

Primitive Function

Initial Condition

General Solution

Worked Examples

Find ∫ (4x3 − 6x + 1/x) dx.

Find ∫ (e2x + cos(x)) dx.

Given f'(x) = 2x − 1 and f(2) = 3, find f(x).

Knowledge Check

Key Concepts Summary

Example: ∫ (5e^x − 3sin(x) + 2x) dx

Example: f'(x) = 3x² and f(1) = 5. Find f(x).

Find ∫ (4x³ − 6x + 1/x) dx.

Find ∫ (e^2x + cos(x)) dx.