Integration by Parts
Reverse of the product rule.
Key idea
Reverse of the product rule.
In depth
∫u dv = uv − ∫v du, from the product rule. Choose u as the part that becomes simpler when differentiated (LIATE: logs, inverse trig, algebraic, trig, exponentials).
Worked example
∫x eˣ dx: u = x, dv = eˣ dx → x eˣ − ∫eˣ dx = x eˣ − eˣ + C.
Formula
Real-world uses
- Finding the total of a product such as distance times decay in physics and probability
Common mistake
Choosing u and dv so the new integral is harder.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Integration by Parts?
Reverse of the product rule. It is a topic at the expert level in Calculus.
What is the formula for Integration by Parts?
The key formula is ∫u dv = uv − ∫v du.
Where is Integration by Parts used in real life?
Real-world examples: Finding the total of a product such as distance times decay in physics and probability.
What should I learn before Integration by Parts?
Make sure you are comfortable with Substitution Rule, the previous topic in Calculus.
What is a common mistake with Integration by Parts?
Choosing u and dv so the new integral is harder.
How can I practise Integration by Parts?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Integration by Parts?
Move on to Trigonometric Integrals next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.