BeginnerTopic 34 of 104Engineering Mathematics
Integration techniques
Substitution, integration by parts and partial fractions evaluate integrals analytically.
In depth
Substitution reverses the chain rule; integration by parts uses ∫u dv = uv-∫v du; partial fractions split rational functions; trigonometric substitution handles √(a²-x²). Choose a technique by the form of the integrand, or use tables and software.
Key points
- LIATE suggests u in parts
- Complete the square for quadratics
- Verify by differentiating
Example∫ x eˣ dx = x eˣ - eˣ + C.
What is Integration techniques?
Substitution, integration by parts and partial fractions evaluate integrals analytically.
Why is it important to understand integration techniques?
Most real integrals are not solved by a table, so these techniques are the tools for turning difficult integrands into standard forms.
When should I use integration by parts?
Use it when the integrand is a product of two functions, one that gets simpler on differentiation and one that is easy to integrate.
What is partial fraction decomposition used for?
It splits a rational function into simpler fractions that integrate easily, and it is standard in Laplace inversion and filter design.
How is Integration techniques used in practice?
Typical applications include integrating a product of a polynomial and an exponential in a transient response, converting a rational function into partial fractions before integrating and changing variables to integrate a circular shape in a stress field.
What should I learn before Integration techniques?
Study "Fundamental theorem of calculus" first (topic 33 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Integration techniques?
LIATE suggests u in parts. Complete the square for quadratics. Verify by differentiating.
Can you give an example of Integration techniques?
∫ x eˣ dx = x eˣ - eˣ + C.
What comes after Integration techniques?
The next topic is "Definite integrals and area", which builds directly on this one.
How difficult is Integration techniques?
It is classed as Beginner in Engineering Mathematics (topic 34 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.