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[Merged by Bors] - feat: compute the integral of sqrt (1 - x ^ 2) #6905

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[Merged by Bors] - feat: compute the integral of sqrt (1 - x ^ 2) #6905

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xroblot Aug 31, 2023
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xroblot Sep 2, 2023
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Merge remote-tracking branch 'origin' into xfr-integral_sqrt
xroblot Sep 5, 2023
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xroblot Sep 5, 2023
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13 changes: 13 additions & 0 deletions Mathlib/Analysis/SpecialFunctions/Integrals.lean
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Expand Up @@ -741,6 +741,19 @@ theorem integral_cos_sq : ∫ x in a..b, cos x ^ 2 = (cos b * sin b - cos a * si
by field_simp [integral_cos_pow, add_sub_assoc]
#align integral_cos_sq integral_cos_sq

theorem integral_sqrt_one_sub_sq : ∫ x in (-1 : ℝ)..1, sqrt (↑1 - x ^ 2) = π / 2 :=
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calc
_ = ∫ x in (-(π / 2))..(π / 2), sqrt (↑1 - sin x ^ 2) * cos x := by
rw [show ↑(-1) = sin (-(π / 2)) by { rw [sin_neg, sin_pi_div_two] }]
nth_rewrite 2 [← sin_pi_div_two]
exact (integral_comp_mul_deriv (fun x _ => hasDerivAt_sin x) continuousOn_cos
(by continuity)).symm
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_ = ∫ x in (-(π / 2))..(π / 2), cos x ^ 2 := by
refine integral_congr_ae (MeasureTheory.ae_of_all _ fun _ h => ?_)
rw [uIoc_of_le (neg_le_self (le_of_lt (half_pos Real.pi_pos))), Set.mem_Ioc] at h
rw [ ← Real.cos_eq_sqrt_one_sub_sin_sq (le_of_lt h.1) h.2, pow_two]
_ = π / 2 := by simp

/-! ### Integral of `sin x ^ m * cos x ^ n` -/


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