Integration by Parts
Evaluate I_n(b)=integral_0^b x^n exp(x) dx using the reduction formula.
Free theory articleEvaluate I_n(b)=integral_0^b x^n exp(x) dx using the reduction formula.
Concept & interpretation
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.
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 bk eb and remainder term k Ik-1 dynamically update the value of Ik 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.