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Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x.
Even Euler does not seem to have written it down explicitly—and certainly it doesn't appear in any of his publications—though he must surely have realized that it follows immediately from his identity [i.e. Euler's formula], e ix = cos x + i sin x. Moreover, it seems to be unknown who first stated the result explicitly
Since cos x is the real part of e ix, we know that ∫ e x cos x d x = Re ∫ e x e i x d x . {\displaystyle \int e^{x}\cos x\,dx=\operatorname {Re} \int e^{x}e^{ix}\,dx.} The integral on the right is easy to evaluate:
For instance, e x can be defined as (+). Or e x can be defined as f x (1), where f x : R → B is the solution to the differential equation df x / dt (t) = x f x (t), with initial condition f x (0) = 1; it follows that f x (t) = e tx for every t in R.
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them.
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The last expression is the logarithmic mean. = ( >) = (>) (the Gaussian integral) = (>) = (, >) (+) = (>)(+ +) = (>)= (>) (see Integral of a Gaussian function
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