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Lição 87 — Integrais trigonométricas e substituição trigonométrica

∫ sinⁿcos^m via identidades e sub u. Substituição trigonométrica para radicais √(a²±x²) e √(x²−a²). Fórmulas de redução de potências.

Used in: Cálculo II (Brasil) · Equiv. Math III japonês · Equiv. Analysis LK alemão · AP Calculus BC (EUA)

\int \sin^n x \cos^m x\, dx \quad \bigg| \quad x = a\sin\theta,\; x = a\tan\theta,\; x = a\sec\theta

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Rigorous notation, full derivation, hypotheses

Identités, motifs et substitutions

Identités fondamentales

Motifs pour $\int \sin^n x \cos^m x\, dx$

Definition· Stratégie selon la parité des exposants

Situation	Stratégie
$n$ impair	Réservez $\sin x\, dx$ ; convertissez $\sin^{n-1}$ via $\sin^2 = 1 - \cos^2$ ; substituez $u = \cos x$
$m$ impair	Réservez $\cos x\, dx$ ; convertissez $\cos^{m-1}$ via $\cos^2 = 1 - \sin^2$ ; substituez $u = \sin x$
Les deux pairs	Utilisez half-angle répétées

Pour $\int \tan^n x \sec^m x\, dx$ :

Situation	Stratégie
$m$ pair	Réservez $\sec^2 x\, dx$ ; convertissez via $\sec^2 = 1 + \tan^2$ ; sub $u = \tan x$
$n$ impair	Réservez $\sec x \tan x\, dx$ ; convertissez via $\tan^2 = \sec^2 - 1$ ; sub $u = \sec x$

"The strategy for integrating a product of powers of sine and cosine depends on the parities of the exponents involved. When one of the exponents is odd, we 'peel off' one factor and use the Pythagorean identity to convert the remaining even power." — OpenStax Calculus Vol. 2, §3.2

Substitution trigonométrique

"The idea behind trigonometric substitution is to replace an expression involving a square root with a trigonometric expression, which is easier to integrate." — APEX Calculus §6.4

Formules de réduction

Exemplos resolvidos

Example— 1· Potência ímpar de seno (básico)

Problema. Calcule $\int \sin^3 x\, dx$ .

Estratégia. Expoente ímpar: reserve $\sin x\, dx$ , converta $\sin^2 = 1 - \cos^2$ , substitua $u = \cos x$ .

Resolução.

$\int \sin^3 x\, dx = \int \sin^2 x \cdot \sin x\, dx = \int (1 - \cos^2 x)\sin x\, dx$ .

Sub $u = \cos x$ , $du = -\sin x\, dx$ :

$-\int (1 - u^2)\, du = -u + u^3/3 + C = -\cos x + \frac{\cos^3 x}{3} + C$ .

Verificação. Derivando: $\sin x - \cos^2 x\sin x = \sin x(1 - \cos^2 x) = \sin^3 x$ . Correto.

Fonte. APEX Calculus §6.3, Example 6.3.1 — CC-BY-NC.

Example— 2· Expoentes pares — half-angle (básico)

Problema. Calcule $\int \sin^2 x \cos^2 x\, dx$ .

Estratégia. Ambos pares: use $\sin x\cos x = \sin(2x)/2$ e depois half-angle.

Resolução.

$\sin^2 x\cos^2 x = (\sin x\cos x)^2 = \frac{\sin^2(2x)}{4} = \frac{1}{4}\cdot\frac{1 - \cos 4x}{2} = \frac{1 - \cos 4x}{8}$ .

$\int \sin^2 x\cos^2 x\, dx = \frac{x}{8} - \frac{\sin 4x}{32} + C.$

Verificação. Derivando: $1/8 - \cos(4x)/8 = (1 - \cos 4x)/8 = \sin^2(2x)/4 = \sin^2 x\cos^2 x$ . Correto.

Fonte. OpenStax Calculus Vol. 2, §3.2, Example 3.12 — CC-BY-NC-SA.

Example— 3· Substituição x = a sin θ para √(a² − x²) (intermediário)

Problema. Calcule $\int \sqrt{4 - x^2}\, dx$ .

Estratégia. Radical $\sqrt{a^2 - x^2}$ com $a = 2$ : sub $x = 2\sin\theta$ .

Resolução.

$dx = 2\cos\theta\, d\theta$ ; $\sqrt{4 - x^2} = \sqrt{4 - 4\sin^2\theta} = 2\cos\theta$ (com $\theta \in [-\pi/2, \pi/2]$ ).

$\int 2\cos\theta \cdot 2\cos\theta\, d\theta = 4\int\cos^2\theta\, d\theta = 4\cdot\frac{\theta + \sin\theta\cos\theta}{2} = 2\theta + 2\sin\theta\cos\theta + C$ .

Triângulo de referência ( $\sin\theta = x/2$ , hipotenusa 2, cateto adjacente $\sqrt{4-x^2}$ ):

$\theta = \arcsin(x/2)$ ; $\cos\theta = \sqrt{4-x^2}/2$ .

Resultado: $2\arcsin(x/2) + 2\cdot\frac{x}{2}\cdot\frac{\sqrt{4-x^2}}{2} + C = 2\arcsin(x/2) + \frac{x\sqrt{4-x^2}}{2} + C$ .

Verificação. Derivando (regra da cadeia): $\frac{2}{\sqrt{4-x^2}} + \frac{\sqrt{4-x^2} - x^2/\sqrt{4-x^2}}{2} = \frac{2}{\sqrt{4-x^2}} + \frac{4 - 2x^2}{2\sqrt{4-x^2}} = \sqrt{4-x^2}$ . Correto.

Fonte. APEX Calculus §6.4, Example 6.4.2 — CC-BY-NC.

Example— 4· Substituição x = a tan θ para √(a² + x²) (intermediário)

Problema. Calcule $\int \dfrac{1}{\sqrt{1 + x^2}}\, dx$ .

Estratégia. Radical $\sqrt{a^2 + x^2}$ com $a = 1$ : sub $x = \tan\theta$ .

Resolução.

$dx = \sec^2\theta\, d\theta$ ; $\sqrt{1+x^2} = \sec\theta$ .

$\int \frac{\sec^2\theta}{\sec\theta}\, d\theta = \int \sec\theta\, d\theta = \ln\lvert\sec\theta + \tan\theta\rvert + C$ .

Triângulo ( $\tan\theta = x$ , hipotenusa $\sqrt{1+x^2}$ ): $\sec\theta = \sqrt{1+x^2}$ , $\tan\theta = x$ .

Resultado: $\ln\lvert x + \sqrt{1+x^2}\rvert + C$ (equivalente a $\text{arcsinh}(x) + C$ ).

Verificação. Derivando $\ln(x + \sqrt{1+x^2})$ : $\frac{1 + x/\sqrt{1+x^2}}{x + \sqrt{1+x^2}} = \frac{(\sqrt{1+x^2}+x)/\sqrt{1+x^2}}{x+\sqrt{1+x^2}} = \frac{1}{\sqrt{1+x^2}}$ . Correto.

Fonte. OpenStax Calculus Vol. 2, §3.3, Example 3.15 — CC-BY-NC-SA.

Example— 5· Fórmula de redução e aplicação (desafio)

Problema. Derive a fórmula de redução $\int \sin^n x\, dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2}x\, dx$ e use-a para calcular $\int_0^{\pi/2}\sin^4 x\, dx$ .

Estratégia. Derivar por partes; aplicar para $n = 4$ e $n = 2$ .

Resolução.

Derivação: $u = \sin^{n-1}x$ , $dv = \sin x\, dx$ . $du = (n-1)\sin^{n-2}x\cos x\, dx$ , $v = -\cos x$ .

$\int\sin^n x\, dx = -\sin^{n-1}x\cos x + (n-1)\int\cos^2 x\sin^{n-2}x\, dx$ .

Substitua $\cos^2 x = 1 - \sin^2 x$ :

$(n-1)\int(1 - \sin^2 x)\sin^{n-2}x\, dx = (n-1)\int\sin^{n-2}x\, dx - (n-1)\int\sin^n x\, dx$ .

Reorganize: $n\int\sin^n x\, dx = -\sin^{n-1}x\cos x + (n-1)\int\sin^{n-2}x\, dx$ . Divida por $n$ .

Aplicação: $n = 4$ : $\int\sin^4 x\, dx = -\frac{\sin^3 x\cos x}{4} + \frac{3}{4}\int\sin^2 x\, dx$ .

$\int\sin^2 x\, dx = x/2 - \sin(2x)/4$ .

Avaliando em $[0, \pi/2]$ : $\int_0^{\pi/2}\sin^4 x\, dx = 0 + \frac{3}{4}\left[\frac{\pi}{4} - 0\right] = \frac{3\pi}{16}$ .

Verificação. Fórmula de Wallis: $\int_0^{\pi/2}\sin^4 x\, dx = \frac{3\cdot 1}{4\cdot 2}\cdot\frac{\pi}{2} = \frac{3\pi}{16}$ . Confirma.

Fonte. APEX Calculus §6.3, Theorem 6.3.1 — CC-BY-NC.

Exercise list

32 exercises · 8 with worked solution (25%)

Application 24Modeling 4Challenge 2Proof 2

Ex. 87.1Application
Calculez $\int \sin^2 x\, dx$ .
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Ex. 87.2Application
Calculez $\int \cos^2 x\, dx$ .
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Ex. 87.3ApplicationAnswer key
Calculez $\int \sin^3 x\, dx$ .
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Ex. 87.4Application
Calculez $\int \cos^3 x\, dx$ .
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Ex. 87.5ApplicationAnswer key
Calculez $\int \sin^4 x\, dx$ .
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Ex. 87.6Application
Calculez $\int \cos^4 x\, dx$ .
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Ex. 87.7Application
Calculez $\int \sin^5 x\, dx$ .
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Ex. 87.8Application
Calculez $\int \tan^2 x\, dx$ .
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Ex. 87.9ApplicationAnswer key
Calculez $\int \sin^2 x \cos^2 x\, dx$ .
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Ex. 87.10ApplicationAnswer key
Calculez $\int \sin^3 x \cos x\, dx$ .
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Ex. 87.11ApplicationAnswer key
Calculez $\int \sin x \cos^3 x\, dx$ .
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Ex. 87.12Application
Calculez $\int \sin^3 x \cos^2 x\, dx$ .
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Ex. 87.13Application
Calculez $\int \sin(3x)\cos(2x)\, dx$ .
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Ex. 87.14Application
Calculez $\int \tan^2 x \sec^2 x\, dx$ .
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Ex. 87.15Application
Calculez $\int \sqrt{1 - x^2}\, dx$ .
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Ex. 87.16Application
Calculez $\int \sqrt{4 - x^2}\, dx$ .
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Ex. 87.17Application
Calculez $\int \dfrac{1}{\sqrt{1 - x^2}}\, dx$ .
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Ex. 87.18ApplicationAnswer key
Calculez $\int \dfrac{1}{\sqrt{9 - x^2}}\, dx$ .
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Ex. 87.19Application
Calculez $\int \dfrac{x^2}{\sqrt{1 - x^2}}\, dx$ .
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Ex. 87.20Application
Calculez $\int \dfrac{1}{1 + x^2}\, dx$ via la substitution $x = \tan\theta$ .
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Ex. 87.21Application
Calculez $\int \dfrac{1}{\sqrt{1 + x^2}}\, dx$ .
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Ex. 87.22Application
Calculez $\int \sqrt{1 + x^2}\, dx$ .
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Ex. 87.23ApplicationAnswer key
Calculez $\int \dfrac{1}{\sqrt{x^2 - 1}}\, dx$ .
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Ex. 87.24Application
Calculez $\int \sqrt{x^2 - 4}\, dx$ .
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Ex. 87.25Modeling
Démontrez que l'aire du cercle de rayon $r$ est $\pi r^2$ en calculant $A = 4\int_0^r \sqrt{r^2 - x^2}\, dx$ .
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Ex. 87.26Modeling
Distribution de Cauchy : vérifiez que $f(x) = \dfrac{1}{\pi(1+x^2)}$ satisfait $\int_{-\infty}^\infty f(x)\, dx = 1$ .
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Ex. 87.27Modeling
Tension efficace (RMS) de $v(t) = V_0\sin(\omega t)$ : calculez $V_{rms} = \sqrt{\frac{1}{T}\int_0^T v^2\, dt}$ où $T = 2\pi/\omega$ .
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Ex. 87.28Modeling
Aire de l'ellipse $x^2/a^2 + y^2/b^2 = 1$ : calculez $A = 4\int_0^a \frac{b}{a}\sqrt{a^2 - x^2}\, dx$ et montrez que $A = \pi ab$ .
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Ex. 87.29ChallengeAnswer key
Calculez $\int \sec^5 x\, dx$ en utilisant la formule de réduction pour $\sec^n$ .
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Ex. 87.30Challenge
Calculez $\int \dfrac{1}{\sin x}\, dx = \int \csc x\, dx$ en utilisant la substitution de Weierstrass $t = \tan(x/2)$ .
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Ex. 87.31Proof
Démonstration. Prouvez la formule de réduction $\int \sin^n x\, dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2}x\, dx$ en utilisant l'intégration par parties.
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Ex. 87.32Proof
Démonstration. Prouvez que $\int \sec x\, dx = \ln\lvert \sec x + \tan x\rvert + C$ en multipliant par $(\sec x + \tan x)/(\sec x + \tan x)$ .
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Fontes

APEX Calculus v5 — Hartman et al. · 2024 · CC-BY-NC · §6.3–6.4. Fonte primária.
Calculus Volume 2 (OpenStax) — OpenStax · 2016 · CC-BY-NC-SA · §3.2–3.3.
Active Calculus 2.0 — Boelkins · 2024 · CC-BY-NC-SA · §5.4–5.5.

Identités, motifs et substitutions

Identités fondamentales

Motifs pour ∫sin⁡nxcos⁡mx dx\int \sin^n x \cos^m x\, dx∫sinnxcosmxdx

Substitution trigonométrique

Formules de réduction

Exemplos resolvidos

Fontes

Motifs pour $\int \sin^n x \cos^m x\, dx$