Formulas Reference Sheet

Useful Summation Formulas:
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Common integrals:

\begin{aligned} (i) & \int \sin x d x = - \cos x + C \\ (ii) & \int \cos x d x = \sin x + C \\ (iii) & \int \sec^{2} x d x = \tan x + C \\ (iv) & \int \sec x \tan x d x = \sec x + C \\ (v) & \int \csc^{2} x d x = - \cot x + C \\ (vi) & \int \csc x \cot x d x = - \csc x + C \\ (vii) & \int \frac{1}{\sqrt{1 - x^{2}}} d x = \arcsin x + C \\ (viii) & \int \frac{- 1}{\sqrt{1 - x^{2}}} d x = \arccos x + C \\ (ix) & \int \frac{1}{1 + x^{2}} d x = \arctan x + C \\ (x) & \int e^{x} d x = e^{x} + C \\ (xi) & \int a^{x} d x = \frac{a^{x}}{\ln a} + C (a > 0, a \neq 1) \\ (xii) & \int \frac{1}{x} d x = \ln | x | + C \\ (xiii) & \int \sec x d x = \ln | \sec x + \tan x | + C \\ (xiv) & \int \csc x d x = - \ln | \csc x + \cot x | + C \\ (xv) & \int \tan x d x = - \ln | \cos x | + C \\ (xvi) & \int \cot x d x = \ln | \sin x | + C \\ (xvi) & \int \cot x d x = \ln | \sin x | + C \end{aligned}

Remember (Pythagorean Identities):

\begin{aligned} (i) & {tan}^{2} (x) + 1 = {sec}^{2} (x) \\ (ii) & \sin^{2} (x) + \cos^{2} (x) = 1 \\ (iii) & \cot^{2} (x) + 1 = \csc^{2} (x) \end{aligned}

(iii) comes from dividing by $\sin (x)$ for (ii)
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They assume you use this domain, don't write it every time! Make it make sense!
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Partial Fraction Decomposition:
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