DPST1013 Chap.9 Integration and Improper Integrals
Integration and Improper Integrals
The definite integral is a limit of signed Riemann sums, and the Fundamental Theorem connects that accumulation to antiderivatives. Substitution and integration by parts reverse familiar derivative structures. Improper integrals replace infinite intervals or unbounded behaviour with limits and require an explicit convergence decision.
What this chapter covers
- 01
Riemann sums
- 02
Definite and indefinite integrals
- 03
Fundamental Theorem of Calculus
- 04
Substitution
- 05
Integration by parts
- 06
Improper integrals and comparison
Worked example · free
Change variables and transform the bounds
- 1Set u = x squared + 3, so du = 2x dx.
- 1Transform the bounds from x = 0 and 1 to u = 3 and 4.
- 1Integrate u to the fourth power to get [u to the fifth / 5] from 3 to 4.
- 1The exact value is (1024 - 243)/5 = 781/5.
Key terms
- Riemann sum
- A finite weighted sum approximating signed accumulation across a partition.
- Antiderivative
- A function whose derivative equals the given integrand.
- Integration by parts
- A product-rule reversal that transfers differentiation from one factor to another.
- Improper integral
- An integral defined by a limit because an endpoint is infinite or the integrand is unbounded.
Integration and Improper Integrals FAQ
Why is a definite integral not always an area?
The definite integral is signed accumulation. Regions below the axis contribute negatively and may cancel positive regions. Geometric area requires finding every crossing, splitting the interval and making each contribution positive.
When should I use substitution?
Look for a composite expression accompanied by a multiple of its derivative. Define the inner expression as the new variable, replace the differential and all remaining occurrences, and transform the bounds for a definite integral.
How do I prove an improper integral converges?
Write its defining limit, evaluate the finite-endpoint integral and take the limit. A finite result proves convergence. If the limit is infinite or fails to exist, state divergence rather than substituting an infinity symbol into an antiderivative.
How can I verify an indefinite integral?
Differentiate the proposed antiderivative and simplify until the original integrand is recovered. The arbitrary constant disappears, as it should. This check exposes missing chain-rule factors, incorrect signs and algebraic simplifications. If the differentiated result differs by a constant multiple, revisit the substitution or integration formula rather than adjusting the answer by intuition.
Domain restrictions may also explain absolute values in logarithmic antiderivatives.
When is integration by parts the natural method?
Use it when the integrand is a product in which differentiating one factor simplifies it and integrating the other remains manageable. Choose the two roles deliberately, write the boundary term for a definite integral and apply the remaining integral with its sign. Repeated applications may be needed for polynomial-exponential or polynomial-trigonometric products. Differentiate the final expression to audit every term.
Why must an improper integral be split at an internal singularity?
Each side of the singular point has an independent limiting process. Write two integrals with separate one-sided limits and require both to converge. Combining them into one symmetric limit can create cancellation that hides divergence. The value of the integrand at the singular point is irrelevant, but its unbounded nearby behaviour determines whether finite signed accumulation exists.
How does the Fundamental Theorem connect derivatives and integrals?
One part states that accumulating a continuous function from a fixed endpoint creates a differentiable function whose derivative is the integrand. The other evaluates a definite integral using any antiderivative. When the upper limit is itself a function, the chain rule supplies its derivative as an extra factor. Keep the accumulation variable separate from the external input to avoid symbol confusion.
What makes a comparison test conclusion valid?
The functions should be non-negative on the relevant tail or near the singularity, and the inequality must point in the useful direction. Being bounded above by a convergent benchmark proves convergence; being bounded below by a divergent benchmark proves divergence. The opposite comparisons do not decide the question. State where the comparison holds and name the known benchmark integral explicitly.
Exam move
Link each integration technique to the derivative rule it reverses. Differentiate every proposed antiderivative as a check. For definite integrals, predict the sign and rough size from the graph, and separate signed accumulation from area. For improper integrals, write the limit before any antiderivative and split at internal singularities.
Learn comparison directions carefully: an upper bound by a convergent function proves convergence, while a lower bound by a divergent function proves divergence. Create a method index based on structure: composite factor, product, rational function, trigonometric identity or improper endpoint. Before calculating a definite integral, sketch enough of the integrand to estimate sign and scale.
Keep substitution bounds in the new variable rather than mixing variables mid-solution. For every convergence exercise, label the dangerous endpoint and write its one-sided limit notation first. Review wrong answers by differentiating them and identifying the exact reverse-rule factor that was lost.
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