Many physics problems encountered in Project Elara require the use of integration to solve (particularly differential equations and physical quantities expressed in terms of definite integrals). These are some common techniques for integration: - Guess the antiderivative and check ("method of inspired guessing") - for reference see [this PDF](http://royalpathtomath.org/docs/Integration%20by%20Guessing.pdf) - Elementary integrals (which are just the derivative rules in reverse) - U-substitution - Integration by parts - Exhaustive integration by parts (D-I method) - Integration by parts cancellation trick - Symmetries: if $f$ is an odd function, then: $ \int_{-a}^a f(x) dx = 0 $ - Multiplying by conjugate: $ \int \frac{1}{1 + \cos x} dx $ $ \int \frac{1}{1 + \cos x} \frac{1 - \cos x}{1 - \cos x} dx $ $ \int \frac{1 - \cos x}{1 - \cos^2 x} dx $ $ \int \frac{1 - \cos x}{\sin^2 x} dx $ $ \int \csc^2 x dx - \int \frac{\cos x}{\sin^2 x} dx $ For first integral - notice that this is just equal to $-\cot x + C$. For the second integral - let $u = \sin x$, then: $ \int \frac{\cos x}{\sin^2 x} dx = \int \frac{1}{u^2} du = -\frac{1}{u} + C $ So the solution is: $ -\cot x + \csc x + C $ - Using trig identities (particularly Pythagorean ones) - e.g. $ \int \sin^2 x dx = \int \frac{1 - \cos (2x)}{2}dx = \frac{1}{2} x - \frac{1}{4} \cos (2x) + C $ - Integration by partial fractions - Long division of integrand - Expanding the integrand - e.g. $ \int (1 + x^2)^2 dx = \int 1+ 2x^2 + x^4 dx = x + \frac{2}{3} x^3 + \frac{1}{5} x^5 + C $ - Cancelling out common factors - Dividing out - e.g. $ \int \frac{1}{\sqrt{16 - 81x^2}} dx = \int \frac{1}{\sqrt{16(1 - (81x^2 / 16))}} = \int \frac{1}{4\sqrt{1 - (9x/4)^2}} $ - Completing the square - Splitting one integral into multiple simpler integrals (fraction-splitting) For fractions: $ \int \frac{1 + x}{x^2} dx = \int \frac{1}{x^2} dx + \int \frac{1}{x} dx = -\frac{1}{x} + \ln |x| + C $ - Trig sub - Integration via geometry (especially for circles, rectangles, and triangles) - Feynmann integration trick (only for definite integrals)