# mathlib3documentation

analysis.special_functions.integrals

# Integration of specific interval integrals #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file contains proofs of the integrals of various specific functions. This includes:

• Integrals of simple functions, such as id, pow, inv, exp, log
• Integrals of some trigonometric functions, such as sin, cos, 1 / (1 + x^2)
• The integral of cos x ^ 2 - sin x ^ 2
• Reduction formulae for the integrals of sin x ^ n and cos x ^ n for n ≥ 2
• The computation of ∫ x in 0..π, sin x ^ n as a product for even and odd n (used in proving the Wallis product for pi)
• Integrals of the form sin x ^ m * cos x ^ n

With these lemmas, many simple integrals can be computed by simp or norm_num. See test/integration.lean for specific examples.

This file also contains some facts about the interval integrability of specific functions.

This file is still being developed.

## Tags #

integrate, integration, integrable, integrability

### Interval integrability #

@[simp]
theorem interval_integral.interval_integrable_pow {a b : } (n : )  :
interval_integrable (λ (x : ), x ^ n) μ a b
theorem interval_integral.interval_integrable_zpow {a b : } {n : } (h : 0 n 0 b) :
interval_integrable (λ (x : ), x ^ n) μ a b
theorem interval_integral.interval_integrable_rpow {a b : } {r : } (h : 0 r 0 b) :
interval_integrable (λ (x : ), x ^ r) μ a b

See interval_integrable_rpow' for a version with a weaker hypothesis on r, but assuming the measure is volume.

See interval_integrable_rpow for a version applying to any locally finite measure, but with a stronger hypothesis on r.

theorem interval_integral.interval_integrable_cpow {a b : } {r : } (h : 0 r.re 0 b) :
interval_integrable (λ (x : ), x ^ r) μ a b

See interval_integrable_cpow' for a version with a weaker hypothesis on r, but assuming the measure is volume.

See interval_integrable_cpow for a version applying to any locally finite measure, but with a stronger hypothesis on r.

@[simp]
theorem interval_integral.interval_integrable_id {a b : }  :
interval_integrable (λ (x : ), x) μ a b
@[simp]
theorem interval_integral.interval_integrable_const {a b : } (c : ) :
interval_integrable (λ (x : ), c) μ a b
theorem interval_integral.interval_integrable_one_div {a b : } {f : } (h : (x : ), x b f x 0) (hf : (set.uIcc a b)) :
interval_integrable (λ (x : ), 1 / f x) μ a b
@[simp]
theorem interval_integral.interval_integrable_inv {a b : } {f : } (h : (x : ), x b f x 0) (hf : (set.uIcc a b)) :
interval_integrable (λ (x : ), (f x)⁻¹) μ a b
@[simp]
@[simp]
theorem interval_integrable.log {a b : } {f : } (hf : (set.uIcc a b)) (h : (x : ), x b f x 0) :
interval_integrable (λ (x : ), real.log (f x)) μ a b
@[simp]
theorem interval_integral.interval_integrable_log {a b : } (h : 0 b) :
@[simp]

### Integrals of the form c * ∫ x in a..b, f (c * x + d)#

@[simp]
theorem interval_integral.mul_integral_comp_mul_right {a b : } {f : } (c : ) :
c * (x : ) in a..b, f (x * c) = (x : ) in a * c..b * c, f x
@[simp]
theorem interval_integral.mul_integral_comp_mul_left {a b : } {f : } (c : ) :
c * (x : ) in a..b, f (c * x) = (x : ) in c * a..c * b, f x
@[simp]
theorem interval_integral.inv_mul_integral_comp_div {a b : } {f : } (c : ) :
c⁻¹ * (x : ) in a..b, f (x / c) = (x : ) in a / c..b / c, f x
@[simp]
theorem interval_integral.mul_integral_comp_mul_add {a b : } {f : } (c d : ) :
c * (x : ) in a..b, f (c * x + d) = (x : ) in c * a + d..c * b + d, f x
@[simp]
theorem interval_integral.mul_integral_comp_add_mul {a b : } {f : } (c d : ) :
c * (x : ) in a..b, f (d + c * x) = (x : ) in d + c * a..d + c * b, f x
@[simp]
theorem interval_integral.inv_mul_integral_comp_div_add {a b : } {f : } (c d : ) :
c⁻¹ * (x : ) in a..b, f (x / c + d) = (x : ) in a / c + d..b / c + d, f x
@[simp]
theorem interval_integral.inv_mul_integral_comp_add_div {a b : } {f : } (c d : ) :
c⁻¹ * (x : ) in a..b, f (d + x / c) = (x : ) in d + a / c..d + b / c, f x
@[simp]
theorem interval_integral.mul_integral_comp_mul_sub {a b : } {f : } (c d : ) :
c * (x : ) in a..b, f (c * x - d) = (x : ) in c * a - d..c * b - d, f x
@[simp]
theorem interval_integral.mul_integral_comp_sub_mul {a b : } {f : } (c d : ) :
c * (x : ) in a..b, f (d - c * x) = (x : ) in d - c * b..d - c * a, f x
@[simp]
theorem interval_integral.inv_mul_integral_comp_div_sub {a b : } {f : } (c d : ) :
c⁻¹ * (x : ) in a..b, f (x / c - d) = (x : ) in a / c - d..b / c - d, f x
@[simp]
theorem interval_integral.inv_mul_integral_comp_sub_div {a b : } {f : } (c d : ) :
c⁻¹ * (x : ) in a..b, f (d - x / c) = (x : ) in d - b / c..d - a / c, f x

### Integrals of simple functions #

theorem integral_cpow {a b : } {r : } (h : -1 < r.re r -1 0 b) :
(x : ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1)
theorem integral_rpow {a b r : } (h : -1 < r r -1 0 b) :
(x : ) in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1)
theorem integral_zpow {a b : } {n : } (h : 0 n n -1 0 b) :
(x : ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1)
@[simp]
theorem integral_pow {a b : } (n : ) :
(x : ) in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1)
theorem integral_pow_abs_sub_uIoc {a b : } (n : ) :
(x : ) in b, |x - a| ^ n = |b - a| ^ (n + 1) / (n + 1)

Integral of |x - a| ^ n over Ι a b. This integral appears in the proof of the Picard-Lindelöf/Cauchy-Lipschitz theorem.

@[simp]
theorem integral_id {a b : } :
(x : ) in a..b, x = (b ^ 2 - a ^ 2) / 2
@[simp]
theorem integral_one {a b : } :
(x : ) in a..b, 1 = b - a
theorem integral_const_on_unit_interval {a b : } :
(x : ) in a..a + 1, b = b
@[simp]
theorem integral_inv {a b : } (h : 0 b) :
(x : ) in a..b, x⁻¹ = real.log (b / a)
@[simp]
theorem integral_inv_of_pos {a b : } (ha : 0 < a) (hb : 0 < b) :
(x : ) in a..b, x⁻¹ = real.log (b / a)
@[simp]
theorem integral_inv_of_neg {a b : } (ha : a < 0) (hb : b < 0) :
(x : ) in a..b, x⁻¹ = real.log (b / a)
theorem integral_one_div {a b : } (h : 0 b) :
(x : ) in a..b, 1 / x = real.log (b / a)
theorem integral_one_div_of_pos {a b : } (ha : 0 < a) (hb : 0 < b) :
(x : ) in a..b, 1 / x = real.log (b / a)
theorem integral_one_div_of_neg {a b : } (ha : a < 0) (hb : b < 0) :
(x : ) in a..b, 1 / x = real.log (b / a)
@[simp]
theorem integral_exp {a b : } :
(x : ) in a..b, rexp x = rexp b - rexp a
theorem integral_exp_mul_complex {a b : } {c : } (hc : c 0) :
(x : ) in a..b, cexp (c * x) = (cexp (c * b) - cexp (c * a)) / c
@[simp]
theorem integral_log {a b : } (h : 0 b) :
(x : ) in a..b, = b * - a * - b + a
@[simp]
theorem integral_log_of_pos {a b : } (ha : 0 < a) (hb : 0 < b) :
(x : ) in a..b, = b * - a * - b + a
@[simp]
theorem integral_log_of_neg {a b : } (ha : a < 0) (hb : b < 0) :
(x : ) in a..b, = b * - a * - b + a
@[simp]
theorem integral_sin {a b : } :
(x : ) in a..b, =
@[simp]
theorem integral_cos {a b : } :
(x : ) in a..b, =
theorem integral_cos_mul_complex {z : } (hz : z 0) (a b : ) :
(x : ) in a..b, complex.cos (z * x) = complex.sin (z * b) / z - complex.sin (z * a) / z
theorem integral_cos_sq_sub_sin_sq {a b : } :
(x : ) in a..b, ^ 2 - ^ 2 = -
@[simp]
theorem integral_inv_one_add_sq {a b : } :
(x : ) in a..b, (1 + x ^ 2)⁻¹ =
theorem integral_one_div_one_add_sq {a b : } :
(x : ) in a..b, 1 / (1 + x ^ 2) =
theorem integral_mul_cpow_one_add_sq {a b : } {t : } (ht : t -1) :
(x : ) in a..b, x * (1 + x ^ 2) ^ t = (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))
theorem integral_mul_rpow_one_add_sq {a b t : } (ht : t -1) :
(x : ) in a..b, x * (1 + x ^ 2) ^ t = (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))

### Integral of sin x ^ n#

theorem integral_sin_pow_aux {a b : } (n : ) :
(x : ) in a..b, ^ (n + 2) = (real.sin a ^ (n + 1) * - ^ (n + 1) * + (n + 1) * (x : ) in a..b, ^ n) - (n + 1) * (x : ) in a..b, ^ (n + 2)
theorem integral_sin_pow {a b : } (n : ) :
(x : ) in a..b, ^ (n + 2) = (real.sin a ^ (n + 1) * - ^ (n + 1) * real.cos b) / (n + 2) + (n + 1) / (n + 2) * (x : ) in a..b, ^ n

The reduction formula for the integral of sin x ^ n for any natural n ≥ 2.

@[simp]
theorem integral_sin_sq {a b : } :
(x : ) in a..b, ^ 2 = (real.sin a * - + b - a) / 2
theorem integral_sin_pow_odd (n : ) :
(x : ) in 0..real.pi, ^ (2 * n + 1) = 2 * (finset.range n).prod (λ (i : ), (2 * i + 2) / (2 * i + 3))
theorem integral_sin_pow_even (n : ) :
(x : ) in 0..real.pi, ^ (2 * n) = real.pi * (finset.range n).prod (λ (i : ), (2 * i + 1) / (2 * i + 2))
theorem integral_sin_pow_pos (n : ) :
0 < (x : ) in 0..real.pi, ^ n
theorem integral_sin_pow_succ_le (n : ) :
(x : ) in 0..real.pi, ^ (n + 1) (x : ) in 0..real.pi, ^ n
theorem integral_sin_pow_antitone  :
antitone (λ (n : ), (x : ) in 0..real.pi, ^ n)

### Integral of cos x ^ n#

theorem integral_cos_pow_aux {a b : } (n : ) :
(x : ) in a..b, ^ (n + 2) = (real.cos b ^ (n + 1) * - ^ (n + 1) * + (n + 1) * (x : ) in a..b, ^ n) - (n + 1) * (x : ) in a..b, ^ (n + 2)
theorem integral_cos_pow {a b : } (n : ) :
(x : ) in a..b, ^ (n + 2) = (real.cos b ^ (n + 1) * - ^ (n + 1) * real.sin a) / (n + 2) + (n + 1) / (n + 2) * (x : ) in a..b, ^ n

The reduction formula for the integral of cos x ^ n for any natural n ≥ 2.

@[simp]
theorem integral_cos_sq {a b : } :
(x : ) in a..b, ^ 2 = (real.cos b * - + b - a) / 2

### Integral of sin x ^ m * cos x ^ n#

theorem integral_sin_pow_mul_cos_pow_odd {a b : } (m n : ) :
(x : ) in a..b, ^ m * ^ (2 * n + 1) = (u : ) in .., u ^ m * (1 - u ^ 2) ^ n

Simplification of the integral of sin x ^ m * cos x ^ n, case n is odd.

@[simp]
theorem integral_sin_mul_cos₁ {a b : } :
(x : ) in a..b, = (real.sin b ^ 2 - ^ 2) / 2

The integral of sin x * cos x, given in terms of sin². See integral_sin_mul_cos₂ below for the integral given in terms of cos².

@[simp]
theorem integral_sin_sq_mul_cos {a b : } :
(x : ) in a..b, ^ 2 * = (real.sin b ^ 3 - ^ 3) / 3
@[simp]
theorem integral_cos_pow_three {a b : } :
(x : ) in a..b, ^ 3 = - (real.sin b ^ 3 - ^ 3) / 3
theorem integral_sin_pow_odd_mul_cos_pow {a b : } (m n : ) :
(x : ) in a..b, ^ (2 * m + 1) * ^ n = (u : ) in .., u ^ n * (1 - u ^ 2) ^ m

Simplification of the integral of sin x ^ m * cos x ^ n, case m is odd.

theorem integral_sin_mul_cos₂ {a b : } :
(x : ) in a..b, = (real.cos a ^ 2 - ^ 2) / 2

The integral of sin x * cos x, given in terms of cos². See integral_sin_mul_cos₁ above for the integral given in terms of sin².

@[simp]
theorem integral_sin_mul_cos_sq {a b : } :
(x : ) in a..b, * ^ 2 = (real.cos a ^ 3 - ^ 3) / 3
@[simp]
theorem integral_sin_pow_three {a b : } :
(x : ) in a..b, ^ 3 = - (real.cos a ^ 3 - ^ 3) / 3
theorem integral_sin_pow_even_mul_cos_pow_even {a b : } (m n : ) :
(x : ) in a..b, ^ (2 * m) * ^ (2 * n) = (x : ) in a..b, ((1 - real.cos (2 * x)) / 2) ^ m * ((1 + real.cos (2 * x)) / 2) ^ n

Simplification of the integral of sin x ^ m * cos x ^ n, case m and n are both even.

@[simp]
theorem integral_sin_sq_mul_cos_sq {a b : } :
(x : ) in a..b, ^ 2 * ^ 2 = (b - a) / 8 - (real.sin (4 * b) - real.sin (4 * a)) / 32