How do I know which method to use?
Ordinary Methods of Integration
\mathrm{\large \int{x^n dx} = } \normalsize \mathrm{\dfrac{x^{n+1}}{n+1} + C}
\\ \: \\ \mathrm{\Large \int{\normalsize \dfrac{1}{x} \ dx} = \large \ln|x| + C}
\\ \: \\ \mathrm{\Large \int{\normalsize \sin(x) \ dx} = \normalsize -\cos(x) + C}
\\ \: \\ \mathrm{\Large \int{\normalsize \cos(x) \ dx} = \normalsize \sin(x) + C}
\\ \: \\ \mathrm{\Large \int{\large e^x \ dx} = \large e^x + C}
\\ \: \\ \mathrm{\Large \int{\normalsize \dfrac{1}{x^2 + a^2} \ dx} = \frac{1}{a} \normalsize \arctan(\dfrac{x}{a}) + C}
Special Methods of Integration
A product of two functions:
\\ \: \\ \mathrm{\Large \int \normalsize f(x)g(x) \: dx}
Use Integration by Parts (the chain rule for integration)
∫u dv = uv – ∫v du
Nested functions:
\\ \: \\ \mathrm{\Large \int \normalsize f(g(x))g'(x) \: dx}
\\ \: \\
Example:
\\ \: \\ \mathrm{\Large \int \normalsize 3x^2 \sin(x^3) \: dx}
\\ \: \\
where
\\ \: \\ \mathrm{u = x^3}
\\ \: \\ \mathrm{du = 3x^2 dx}
A fraction with a polynomial in the denominator that can be factored:
\\ \: \\ \mathrm{\Large \int \normalsize \dfrac{f(x)}{g(x)} \: dx}
\\ \: \\
Example:
\\ \: \\ \mathrm{\Large \int \normalsize \dfrac{3x + 2}{x^2 - x - 6} \: dx}
A function with a square root of a quadratic in one of these three forms:
\\ \: \\ \mathrm{\Large \int \normalsize \sqrt{a^2 - x^2} \: dx}
\\ \: \\ \mathrm{\Large \int \normalsize \sqrt{x^2 + a^2} \: dx}
\\ \: \\ \mathrm{\Large \int \normalsize \sqrt{x^2 - a^2} \: dx}
\\ \: \\
Example:
\\ \: \\ \mathrm{\LARGE \int \normalsize \dfrac{1}{\sqrt{x^2 + 25}} \: dx}
\\ \: \\ \mathrm{\LARGE \int \normalsize \dfrac{1}{\sqrt{4x^2 + 9}} \: dx}
\\ \: \\
Or, a bit harder:
\\ \: \\ \mathrm{\LARGE \int \normalsize \dfrac{1}{\sqrt{x^2 + 10x + 26}} \: dx}
\\ \: \\ \mathrm{= \LARGE \int \normalsize \dfrac{1}{\sqrt{(x^2 + 10x + 25) + 1}} \: dx}
\\ \: \\ \mathrm{= \LARGE \int \normalsize \dfrac{1}{\sqrt{(x^2 + 10x + 25) + 1}} \: dx}
\\ \: \\ \mathrm{= \LARGE \int \normalsize \dfrac{1}{\sqrt{(x + 5)^2 + 1}} \: dx}
Use Integration by Trigonometric Substitution
You may need to factor or complete the square to force the quadratic into this format.
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