2WBB0Calculus I

2WBB0 · Topic 07

Integration techniques and the FTC

Substitution, parts, partial fractions, improper integrals, and differentiation of integrals.

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What you need to be able to do

  • Choose substitution or integration by parts from the integrand structure.
  • Divide and decompose rational functions correctly.
  • Handle improper bounds and variable limits with explicit limits and chain factors.
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1. Substitution versus integration by parts

Substitution reverses the chain rule: look for an inner expression and a multiple of its derivative. Integration by parts reverses the product rule and is suited to products where differentiating one factor simplifies it.

Logarithms and inverse trigonometric functions usually become u. Polynomial times exponential or trigonometric functions often need repeated parts.

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Exam method

  • For substitution, name u and rewrite the entire integral in u before integrating.
  • For parts, choose u to become simpler when differentiated.
  • Keep boundary values consistent: either change the bounds or substitute back.
  • Differentiate the final antiderivative for a quick check.
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Substitution versus integration by parts: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

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Substitution versus integration by partsSelf-review

Worked example 1

Compute ∫19ln⁡xx dx.\int_1^9\frac{\ln x}{\sqrt x}\,dx.
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2. Polynomial division and partial fractions

Partial fractions apply to a proper rational function. If the numerator degree is not smaller, long division comes first.

The denominator factorization dictates the decomposition. Repeated linear factors need every power; irreducible quadratics need a linear numerator.

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Exam method

  • Compare degrees and divide if necessary.
  • Factor the denominator over the reals.
  • Write one term for every required factor and solve coefficients.
  • Integrate logarithmic and arctangent pieces, preserving absolute values.
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Polynomial division and partial fractions: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

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Polynomial division and partial fractionsSelf-review

Worked example 1

Set up the integration of ∫x3−4x−10x2−x−6 dx.\int\frac{x^3-4x-10}{x^2-x-6}\,dx.
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3. Improper integrals and variable bounds

An infinite bound is notation for a limit. Convergence is a conclusion after evaluating that limit, not an assumption.

For variable bounds, the Fundamental Theorem evaluates the integrand at each moving endpoint and the chain rule contributes the endpoint derivative.

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Exam method

  • Replace every infinite or singular endpoint by a limit.
  • Integrate over the finite interval first.
  • Evaluate the limit and explicitly state convergence or divergence.
  • For moving bounds, apply upper contribution minus lower contribution.
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Improper integrals and variable bounds: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

Question 1/2 · 0 reviews recorded
Improper integrals and variable boundsSelf-review

Worked example 1

Evaluate ∫−∞∞ex1+e2x dx.\int_{-\infty}^{\infty}\frac{e^x}{1+e^{2x}}\,dx.
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Formulas used in this topic

  • ∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du
  • ∫f′(x)f(x) dx=ln⁡∣f(x)∣+C\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C
  • ddx∫a(x)b(x)g(t) dt=g(b(x))b′(x)−g(a(x))a′(x)\frac d{dx}\int_{a(x)}^{b(x)}g(t)\,dt=g(b(x))b'(x)-g(a(x))a'(x)
  • ∫a∞f(x) dx=lim⁡R→∞∫aRf(x) dx\int_a^\infty f(x)\,dx=\lim_{R\to\infty}\int_a^R f(x)\,dx
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