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Wize University Calculus 2 Textbook > Integration Techniques

Mixed Integration Questions

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Strategies for Integration


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Strategies

1. Try to simplify if possible (expand brackets that are linear or even quadratic), split up fractions

2. Look for obvious u-substitution (Is there a function wrapped inside another? Does the derivative of one part show up elsewhere?)

3. If we have a product of different function types, try integration by parts

4. If we have a product of trig functions, try trig integrals

5. If we have an x2x^2, try trig sub
  • sometimes you might also have to try completing the square
6. If you have a rational functions, try to use partial fractions (Are there only linear and quadratic terms in the denominator?)

7. Sometimes if we have a root (...x...n\sqrt[n]{...x...}), try rationalizing substitution (see example that follows)


*Keep trying, sometimes you need to use a technique more than once, or you might have to combine techniques.

Practice Question

Evaluate ∫3x−x+2dx{\displaystyle \int}\frac{3}{x-\sqrt{x+2}}dx

Practice Question

Evauate ∫arctan⁡(x)  ⁣dx{\displaystyle\int}\arctan(\sqrt{x})\de{x}.
Hint:
  • Try doing a u-substitution first with u=xu=\sqrt{x}

Practice Question

Evaluate ∫1x2+1dx\displaystyle \int_{ }^{ }\frac{1}{\sqrt{x^2+1}}dx

Practice Question

Evaluate ∫1x4+2x2+1dx\displaystyle\int_{ }^{ }\frac{1}{x^4+2x^2+1}dx

Practice Question

Evaluate ∫x+1x2−2x+6dx\displaystyle\int_{ }^{ }\frac{x+1}{x^2-2x+6}dx

Practice Question

Evaluate ∫1(x2−6x+11)2  ⁣dx{\displaystyle\int} \frac{1}{(x^2-6x+11)^2}\de{x}.