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CIE A-Level Maths Study Notes

2.5.1 Advanced Integration Techniques

Integration, the inverse operation to differentiation, is essential, enabling students to solve a broad spectrum of problems. This section focuses on extending basic integration methods to cover more complex functions such as eax+be^{ax+b}, 1ax+b\frac{1}{ax+b}, sin(ax+b)\sin(ax+b), cos(ax+b)\cos(ax+b), sec2(ax+b)\sec^2(ax+b), and 1ax+b\frac{1}{\sqrt{ax+b}}.

Techniques for Advanced Integration

1. Integration of Exponential Functions

For an exponential function eax+be^{ax+b}, the integral is

eax+bdx=eax+ba+C\int e^{ax+b} \, dx = \frac{e^{ax+b}}{a} + C


e2x+1dx=e2x+12+C\int e^{2x+1} \, dx = \frac{e^{2x+1}}{2} + C

2. Integration of Rational Functions

Integrating a rational function 1ax+b\frac{1}{ax+b}, we find:

1ax+bdx=1alnax+b+C\int \frac{1}{ax+b} \, dx = \frac{1}{a} \ln |ax+b| + C


13x+4dx=13ln3x+4+C\int \frac{1}{3x+4} \, dx = \frac{1}{3} \ln |3x+4| + C

3. Integration of Trigonometric Functions

The integrals for trigonometric functions sin(ax+b)\sin(ax+b) and cos(ax+b)\cos(ax+b)are:

sin(ax+b)dx=1acos(ax+b)+C\int \sin(ax+b) \, dx = -\frac{1}{a} \cos(ax+b) + Ccos(ax+b)dx=1asin(ax+b)+C\int \cos(ax+b) \, dx = \frac{1}{a} \sin(ax+b) + C


sin(2x+1)dx=12cos(2x+1)+C\int \sin(2x+1) \, dx = -\frac{1}{2} \cos(2x+1) + Ccos(2x+1)dx=12sin(2x+1)+C\int \cos(2x+1) \, dx = \frac{1}{2} \sin(2x+1) + C

4. Integration of Secant Squared Function

The integral of sec2(ax+b)\sec^2(ax+b) is:

sec2(ax+b)dx=1atan(ax+b)+C\int \sec^2(ax+b) \, dx = \frac{1}{a} \tan(ax+b) + C


sec2(3x+2)dx=13tan(3x+2)+C\int \sec^2(3x+2) \, dx = \frac{1}{3} \tan(3x+2) + C

5. Integration of Radical Functions

For 1ax+b\frac{1}{\sqrt{ax+b}}, the correct integral involves an inverse sine function due to the square root in the denominator:

1ax+bdx=2asin1(axb)+C\int \frac{1}{\sqrt{ax+b}} \, dx = \frac{2}{\sqrt{a}} \sin^{-1}\left(\frac{\sqrt{a}x}{\sqrt{b}}\right) + C


12x+3dx=22sin1(2x3)+C\int \frac{1}{\sqrt{2x+3}} \, dx = \frac{2}{\sqrt{2}} \sin^{-1}\left(\frac{\sqrt{2}x}{\sqrt{3}}\right) + C
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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