integral_sin — Mathlib

English

For nonzero c ∈ ℂ, ∫_a^b exp(c x) dx = (exp(c b) − exp(c a))/c.

Русский

Пусть c ∈ ℂ, c ≠ 0. Тогда ∫_a^b e^{c x} dx = (e^{c b} − e^{c a})/c.

LaTeX

$$$\forall a,b \in \mathbb{R},\ \forall c \in \mathbb{C},\ c \neq 0 \Rightarrow \displaystyle \int_a^b e^{c x} \, dx = \frac{e^{c b} - e^{c a}}{c}$$$

Lean4

@[simp]
theorem integral_sin : ∫ x in a..b, sin x = cos a - cos b :=
  by
  rw [integral_deriv_eq_sub' fun x => -cos x]
  · ring
  · simp
  · simp only [differentiableAt_fun_neg_iff, differentiableAt_cos, implies_true]
  · exact continuousOn_sin

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