Differentiating Under the Integral
Swap derivative and integral under conditions.
Key idea
Swap derivative and integral under conditions.
In depth
d/dt ∫ₐᵇ f(x, t) dx = ∫ₐᵇ ∂f/∂t dx for fixed limits (under mild conditions). With variable limits there are extra boundary terms. It solves integrals by introducing a parameter.
Worked example
I(t) = ∫₀¹ xᵗ dx = 1/(t + 1); I′(t) = ∫ xᵗ ln x dx = −1/(t + 1)².
Real-world uses
- Deriving formulas for average quantities that depend on a changing parameter in statistics and physics
Common mistake
Forgetting the boundary terms when the limits depend on t.
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 Differentiating Under the Integral?
Swap derivative and integral under conditions. It is a topic at the expert level in Calculus.
Where is Differentiating Under the Integral used in real life?
Real-world examples: Deriving formulas for average quantities that depend on a changing parameter in statistics and physics.
What should I learn before Differentiating Under the Integral?
Make sure you are comfortable with Divergence Theorem, the previous topic in Calculus.
What is a common mistake with Differentiating Under the Integral?
Forgetting the boundary terms when the limits depend on t.
How can I practise Differentiating Under the Integral?
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 Differentiating Under the Integral?
Move on to Gamma Function 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.