Basic Integration (AI SL)
Integration reverses differentiation: given a derivative, work backwards to find the original function. For polynomial terms this is a single rule applied to every term, plus a constant that a boundary condition can pin down. This page covers the power rule for integration, finding a curve from its gradient, and the mistakes that cost the most marks. It's part of the broader Integration topic.
41 questions on this sub-topic.
The power rule for integration
Covered under IB syllabus reference SL5.5: introduction to integration as anti-differentiation of functions of the form \(f(x)=ax^n+bx^{n-1}+\cdots\), with a boundary condition to determine the constant term.
Power rule for integration
\[\int ax^n\,dx=\dfrac{ax^{n+1}}{n+1}+C,\ n\neq-1\]
Increase the exponent by one, then divide by the new exponent. Not in the formula booklet - it's treated as prior knowledge.
Constant of integration
Every indefinite integral needs a "+C" - it represents the unknown vertical shift of the original function, since a constant vanishes on differentiation.
A boundary condition, such as a point the curve passes through, lets you solve for the exact value of \(C\).
Need definite integrals or area under a curve instead? See Definite Integrals and Area.
Worked examples
Find \(\displaystyle\int (6x^2 - 4x + 5)\,dx.\)
Worked solution
Increase each power by one and divide by the new power. M1
\(\int \(6x^2 - 4x + 5\)\,dx\) A1
\(= 2x^3 - 2x^2 + 5x + C.\) A1
A curve has gradient \(\dfrac{dy}{dx} = 3x^2 - 2\) and passes through \((1, 4).\) Find the equation of the curve.
Worked solution
\(y = x^3 - 2x + C.\) M1 A1
\(4 = 1 - 2 + C \Rightarrow C\) M1 \(= 5.\) A1 So \(y = x^3 - 2x + 5.\) A1
Common mistakes
- Increasing the exponent but forgetting to divide by the new value. \(\displaystyle\int x^3\,dx=\dfrac{x^4}{4}+C\), not \(x^4+C\) - both steps of the power rule are needed.
- Losing the "+C" and then having nothing to solve for. If a boundary condition is given, it exists specifically to pin down \(C\) - dropping the constant early means there's no unknown left to substitute into.
- Integrating a constant term as if it disappears. A lone number like \(5\) is really \(5x^0\), so \(\displaystyle\int 5\,dx = 5x + C\), not just \(5\) - it's easy to skip constant terms when integrating a longer expression.
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44 basic integration questions, marked instantly like the real exam.
Quick answers
What is the power rule for integration?
\(\displaystyle\int ax^n\,dx = \dfrac{ax^{n+1}}{n+1}+C\), for \(n\neq-1\): increase the exponent by one, then divide by the new exponent.
Why does an indefinite integral need a +C?
Because a constant vanishes when you differentiate, any vertical shift of the antiderivative has the same derivative - the \(+C\) represents that unknown shift. A boundary condition (a known point on the curve) lets you find its exact value.