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Integral and the usage in mensuration

Their area under a curve can be obtained by using the integral with a certain bounded interval. Their antiderivatives are: ∫ sin (x) d x = − cos (x) + C ∫ cos (x) d x = sin (x) + C, {\displaystyle \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(x)+C,} where C {\displaystyle C} denotes the constant of integration. These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given interval. For example, the arc length of the sine curve between 0 {\displaystyle 0} and t {\displaystyle t} is ∫ 0 t 1 + cos 2 (x) d x = 2 E (t, 1 2), {\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E} \left(t,{\frac {1}{\sqrt {2}}}\right),} where E (φ, k) {\displaystyle \operatorname {E} (\varphi,k)} is the incomplete elliptic integral of the second kind with modulus k {\displaystyle k}. It cannot be expressed using elementary functions. In the case of a full period, its arc length is L = 4 2 π 3 Γ (1 / 4) 2 + Γ (1 / 4) 2 2 π = 2 π ϖ + 2 ϖ ≈ 7.6404 … {\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots } where Γ {\displaystyle \Gamma } is the gamma function and ϖ {\displaystyle \varpi } is the lemniscate constant.