Integration by parts, trigonometric substitution, partial fractions, improper integrals and numerical integration.
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- Formula for integration by parts
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integral of u dv equals uv minus integral of v du- LIATE rule in integration by parts
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Choose u in order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential- When is integration by parts useful?
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When the integrand is a product of two functions, especially when one differentiates to something simpler- What does tabular integration streamline?
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Repeated integration by parts, organizing u derivatives and dv integrals in columns- Integral of x sin(x) dx
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-x cos(x) + sin(x) + C- Integral of x times e to the x power dx
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e^x(x - 1) + C- Integral of x cos(x) dx
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x sin(x) + cos(x) + C- Integral of ln(x) dx
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x ln(x) - x + C- How to resolve cyclic integration by parts
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Treat the integral as a variable when the same integral reappears, then solve algebraically- Substitution for integrals containing sqrt(a squared minus x squared)
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x = a sin(theta), dx = a cos(theta) d(theta)- Substitution for integrals containing sqrt(a squared plus x squared)
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x = a tan(theta), dx = a sec squared(theta) d(theta)- Substitution for integrals containing sqrt(x squared minus a squared)
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x = a sec(theta), dx = a sec(theta) tan(theta) d(theta)- How to use a right triangle in trigonometric substitution
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Draw the triangle to convert back to the original variable x from trig functions- Integral of 1 over sqrt(1 minus x squared) dx
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arcsin(x) + C- Integral of 1 over (1 plus x squared) dx
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arctan(x) + C
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