Standard advanced integrals

Objective. Use logarithmic and inverse-trigonometric forms.

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Learn: the key idea

Standard advanced integrals include results beyond polynomials, such as ∫1/x dx = ln|x| + c, ∫1/(1+x²) dx = arctan x + c, and ∫1/√(1−x²) dx = arcsin x + c. These arise because they are the reverse of known derivatives of ln x, arctan x and arcsin x. For example, ∫ 1/(1+4x²) dx uses the arctan pattern scaled by a factor, giving (1/2)arctan(2x) + c. Recognising the denominator's shape tells you which inverse function to use.

Worked example

Find the standard integral ∫ 1/√(4−x²) dx.

1 $Recognise the form 1/\sqrt(a²-x²), which integrates to arcsin(x/a)$
2 $Identify a² = 4, so a = 2$
3 $Apply the standard result arcsin(x/a) + c$
4 $Write the final answer arcsin(x/2) + c$

Answer: arcsin(x/2) + c

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Common mistake. Students often integrate 1/(1+x²) as ln(1+x²) by confusing it with the 1/x pattern. Always match the denominator's exact form: 1+x² signals arctan x, while a single power of x signals a natural logarithm.

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