Advanced Integration Techniques (2.5.1) | CIE A-Level Maths Notes

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 $e^{ax+b}$ , $\frac{1}{ax+b}$ , $\sin(ax+b)$ , $\cos(ax+b)$ , $\sec^2(ax+b)$ , and $\frac{1}{\sqrt{ax+b}}$ .

Techniques for Advanced Integration

1. Integration of Exponential Functions

For an exponential function $e^{ax+b}$ , the integral is

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

Example:

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

2. Integration of Rational Functions

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

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

Example:

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

3. Integration of Trigonometric Functions

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

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

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

Examples:

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

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

4. Integration of Secant Squared Function

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

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

Example:

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

5. Integration of Radical Functions

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

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

Example:

\int \frac{1}{\sqrt{2x+3}} \, dx = \frac{2}{\sqrt{2}} \sin^{-1}\left(\frac{\sqrt{2}x}{\sqrt{3}}\right) + C

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.

CIE A-Level Maths Study Notes

2.5.1 Advanced Integration Techniques

Techniques for Advanced Integration

1. Integration of Exponential Functions

Example:

2. Integration of Rational Functions

Example:

3. Integration of Trigonometric Functions

Examples:

4. Integration of Secant Squared Function

Example:

5. Integration of Radical Functions

Example:

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