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A2 Level · Pure Mathematics 3 WMA13

Further Integration

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Further Integration

Master reverse chain rule, special fractions, and trigonometric integration for Edexcel IAL Pure 3

What You'll Learn

Integrating exponentials, logarithms, and trig functions
Recognizing and using the reverse chain rule
Integrating fractions where the numerator is the derivative of the denominator
Simplifying trig expressions before integrating
Adjusting and compensating for constants
When to use the formula booklet vs. memorized rules

Integrating Other Functions

Integration is the reverse of differentiation. If you know what the derivative of a function is, you can work backwards to find the integral. The key is to recognize common function families and remember that you must always add the constant of integration + c for indefinite integrals.

Why +c? Because the derivative of a constant is zero. When you integrate, you're asking "what function has this derivative?" The answer is infinitely many functions — all differing by a constant. That's why ∫ f(x) dx = F(x) + c, where c is called the constant of integration. (For definite integrals, the c cancels out when you evaluate at the bounds, so you don't write it.)

Exponential Functions: e^x and e^kx

The beauty of e^x is that it's its own derivative. This makes integration almost as easy as differentiation.

∫ e^x dx = e^x + c

When there's a constant multiplier in the exponent:

∫ e^kx dx = (1/k) e^kx + c

Why the 1/k? Because when you differentiate e^kx using the chain rule, you get k · e^kx. So to integrate, you need to divide by k to undo that multiplication.

Example: ∫ 5e^3x dx

Step 1: Recognize this is an exponential of the form k · e^mx where k = 5 and m = 3.

Step 2: Use the formula: ∫ 5e^3x dx = 5 · (1/3) e^3x + c

Step 3: Simplify: = (5/3) e^3x + c

Check: Differentiate your answer: d/dx[(5/3)e^3x] = (5/3) · 3 · e^3x = 5e^3x ✓

Try This:

Find: ∫ 8e^2x dx

Integrating 1/x and Logarithms

The integral of 1/x is not straightforward using the power rule (because the power is −1, which breaks the formula). Instead, it gives you a logarithm:

∫ (1/x) dx = ln|x| + c

Critical: The modulus (absolute value) The vertical bars |x| are not optional. Because ln is only defined for positive numbers, and because you might be integrating from negative bounds, you must write ln|x|. This ensures the argument of the logarithm is always positive.

A Common Trap: Writing ∫ (1/x) dx = ln(x) + c instead of ln|x| + c will cost you marks. Always use the modulus.

Example: Definite Integral with 1/x

Evaluate ∫_-3^-1 (1/x) dx

Step 1: Find the antiderivative: ∫ (1/x) dx = ln|x| + c

Step 2: Evaluate at the bounds:
[ln|x|]_-3^-1 = ln|-1| − ln|-3| = ln(1) − ln(3) = 0 − ln(3) = −ln(3)

Trigonometric Functions: sin x and cos x

Sine and cosine are closely linked through differentiation and integration. Remember that d/dx[sin x] = cos x and d/dx[cos x] = −sin x. So when you reverse these:

∫ sin x dx = −cos x + c
∫ cos x dx = sin x + c

Watch the minus sign! −

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Also in the full note

The Reverse Chain Rule
Integrating f'(x)/f(x)
Integrating with Trigonometric Identities
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
Integrating tan x
What is the Chain Rule?

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Further Integration

Further Integration

What You'll Learn

Integrating Other Functions

Exponential Functions: ex and ekx

Example: ∫ 5e3x dx

Try This:

Integrating 1/x and Logarithms

Example: Definite Integral with 1/x

Trigonometric Functions: sin x and cos x

More Pure Mathematics 3 topics

Exponential Functions: e^x and e^kx

Example: ∫ 5e^3x dx