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The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals.
At this point we can either integrate directly, or we can first change the integrand to 2 cos 6x − 4 cos 4x + 2 cos 2x and continue from there. Either method gives Either method gives ∫ sin 2 x cos 4 x d x = − 1 24 sin 6 x + 1 8 sin 4 x − 1 8 sin 2 x + C . {\displaystyle \int \sin ^{2}x\cos 4x\,dx=-{\frac {1}{24 ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
3.1 Integrals of hyperbolic tangent, cotangent, secant, cosecant functions 3.2 Integrals involving hyperbolic sine and cosine functions 3.3 Integrals involving hyperbolic and trigonometric functions
In the integral , we may use = , = , = . Then, = = () = = = + = +. The above step requires that > and > We can choose to be the principal root of , and impose the restriction / < < / by using the inverse sine function.
Leonhard Euler used it to evaluate the integral / (+ ) in his 1768 integral calculus textbook, [3] and Adrien-Marie Legendre described the general method in 1817. [ 4 ] The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [ 5 ]
In general, similar integrals have value π / 2 whenever the numbers 3, 5, 7… are replaced by positive real numbers such that the sum of their reciprocals is less than 1. In the example above, 1 / 3 + 1 / 5 + … + 1 / 13 < 1, but 1 / 3 + 1 / 5 + … + 1 / 15 > 1.
Plot of Si(x) for 0 ≤ x ≤ 8π. Plot of the cosine integral function Ci(z) in the complex plane from −2 − 2i to 2 + 2i. The different sine integral definitions are = = .