Integration by Parts

Integration by parts is a powerful technique in calculus that is derived from the product rule for differentiation. It allows us to evaluate the integral of a product of functions by transforming it into another, hopefully simpler, integral.

Worked Example

Let's evaluate the definite integral I₂(1) = ∫₀¹ x² e^x dx. We use the formula: ∫ u dv = uv - ∫ v du.

Let u = x² and dv = e^x dx.
Then du = 2x dx and v = e^x.
Apply the formula:
I₂ = [ x² e^x ]₀¹ - ∫₀¹ 2x e^x dx
I₂ = (1² e¹ - 0) - 2 ∫₀¹ x e^x dx
I₂ = e - 2 I₁
Now, calculate I₁(1):
Let u = x and dv = e^x dx.
Then du = dx and v = e^x.
Apply the formula again:
I₁ = [ x e^x ]₀¹ - ∫₀¹ e^x dx
I₁ = (1 · e¹ - 0) - [ e^x ]₀¹
I₁ = e - (e¹ - e⁰) = e - e + 1 = 1
Substitute I₁ back into I₂:
I₂ = e - 2(1) = e - 2 ≈ 0.718281828459

Interactive Experiment

You can explore this reduction process using the Integration by Parts module.

Set Power n to 2.
Set Upper bound b to 1.
Observe the full reduction ledger. It shows the calculated values for I₂, I₁, and I₀ sequentially.
Note how the boundary term b^k e^b and remainder term k I_k-1 dynamically update the value of I_k for each step.
Set Power n to 0 and Upper bound b to 1. The single base row is e − 1, approximately 1.71828182846; no negative power is evaluated.
Set Upper bound b to 0, then set Power n to 3. Every displayed integral is exactly 0 and the ledger still ends at I₀.
Keep Power n = 3 and set Upper bound b to 2. Read 2e² + 6, approximately 20.7781121979.
Clear Upper bound b to see its error and hidden reduction results. Restore 1, pan the curve, then choose Reset view; the two input values remain unchanged.

Model Limitations

This experiment is specifically bounded to integer powers n ∈ [0, 3] and decimal upper bounds b ∈ [0, 2]. The ledger shows the exact recursive reduction identity for the integral ∫₀^b xⁿ e^x dx. Numerically, positive integrated exponential-series terms avoid cancellation for small positive b; the displayed values are rounded, not symbolic exact decimals. The graph adapts to the current upper bound, so its pixel scale changes with b. Non-integer powers and upper bounds outside the specified domain are outside this experiment; no claim about their general integrability is made. Additionally, values are rendered using floating-point approximations with limited significant digits.

References

The mathematical formulas and approaches in this module are backed by the following resources:

OpenStax Calculus Volume 2: 3.1 Integration by Parts — Derivation of the integration by parts formula from the product rule and its application to definite integrals.

Integration by Parts