Radford Mathematics IB Mathematics resources • AA HL

IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Standard Integrals

Recognise standard antiderivatives and integrate linear composites with correct scale factors.

AA HL · SL 5.10

Learning goal

Integrate standard powers, trigonometric, exponential and reciprocal functions.

Syllabus link

AA HL: SL 5.10 · Topic 5 Calculus.

Big idea

Integration is reverse differentiation: each standard integral should be checked mentally by differentiating the result.

Key relationship

\(\int x^n dx=\frac{x^{n+1}}{n+1}+C\;(n\ne-1)\)

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Core idea

Integration is reverse differentiation: each standard integral should be checked mentally by differentiating the result.

Integrate standard powers, trigonometric, exponential and reciprocal functions.

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Key results and method

\[\int x^n dx=\frac{x^{n+1}}{n+1}+C\;(n\ne-1)\]
\[\int\frac1x dx=\ln|x|+C\]
\[\int e^x dx=e^x+C\]
\[\int\sin x dx=-\cos x+C\]
\[\int\cos x dx=\sin x+C\]
  1. Identify the standard form.
  2. For \(f(ax+b)\), account for the derivative \(a\).
  3. Include \(+C\).
  4. Differentiate the result as a quick check.
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Worked example

Question

Evaluate \(\int e^{3x+1}\,dx\).

Solution
  1. The inner derivative of \(3x+1\) is \(3\).
  2. Divide by 3 when reversing the chain rule.
Answer: \(\frac13e^{3x+1}+C\).
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Practice

Try these questions before moving on.

  1. Integrate \(\cos(4x-1)\).
  2. Integrate \(x^{-1}\).
  3. Integrate a sum containing powers and exponentials.

Search the worksheet library for more practice →

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