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Lesson 87 — Trigonometric integrals and trigonometric substitution

∫ sinⁿcos^m via identities and sub u. Trigonometric substitution for radicals √(a²±x²) and √(x²−a²). Power reduction formulas.

Used in: Calculus II (Brazil) · Equiv. Math III Japanese · Equiv. Analysis LK German · AP Calculus BC (USA)

\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

Identities, patterns, and substitutions

Fundamental identities

Patterns for $\int \sin^n x \cos^m x\, dx$

Definition· Strategy by exponent parity

Situation	Strategy
$n$ odd	Reserve $\sin x\, dx$ ; convert $\sin^{n-1}$ via $\sin^2 = 1 - \cos^2$ ; substitute $u = \cos x$
$m$ odd	Reserve $\cos x\, dx$ ; convert $\cos^{m-1}$ via $\cos^2 = 1 - \sin^2$ ; substitute $u = \sin x$
Both even	Use half-angle repeatedly

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

Situation	Strategy
$m$ even	Reserve $\sec^2 x\, dx$ ; convert via $\sec^2 = 1 + \tan^2$ ; sub $u = \tan x$
$n$ odd	Reserve $\sec x \tan x\, dx$ ; convert 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

Trigonometric substitution

"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

Reduction formulas

Worked examples

Example— 1· Odd power of sine (basic)

Problem. Compute $\int \sin^3 x\, dx$ .

Strategy. Odd exponent: reserve $\sin x\, dx$ , convert $\sin^2 = 1 - \cos^2$ , substitute $u = \cos x$ .

Solution.

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

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

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

Check. Differentiating: $\sin x - \cos^2 x\sin x = \sin x(1 - \cos^2 x) = \sin^3 x$ . Correct.

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

Example— 2· Even exponents — half-angle (basic)

Problem. Compute $\int \sin^2 x \cos^2 x\, dx$ .

Strategy. Both even: use $\sin x\cos x = \sin(2x)/2$ and then half-angle.

Solution.

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

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

Check. Differentiating: $1/8 - \cos(4x)/8 = (1 - \cos 4x)/8 = \sin^2(2x)/4 = \sin^2 x\cos^2 x$ . Correct.

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

Example— 3· Substitution x = a sin θ for √(a² − x²) (intermediate)

Problem. Compute $\int \sqrt{4 - x^2}\, dx$ .

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

Solution.

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

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

Reference triangle ( $\sin\theta = x/2$ , hypotenuse 2, adjacent side $\sqrt{4-x^2}$ ):

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

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

Check. Differentiating (chain rule): $\frac{2}{\sqrt{4-x^2}} + \frac{\sqrt{4-x^2} - x^2/\sqrt{4-x^2}}{2} = \frac{2}{\sqrt{4-x^2}} + \frac{4 - 2x^2}{2\sqrt{4-x^2}} = \sqrt{4-x^2}$ . Correct.

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

Example— 4· Substitution x = a tan θ for √(a² + x²) (intermediate)

Problem. Compute $\int \dfrac{1}{\sqrt{1 + x^2}}\, dx$ .

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

Solution.

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

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

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

Result: $\ln\lvert x + \sqrt{1+x^2}\rvert + C$ (equivalent to $\text{arcsinh}(x) + C$ ).

Check. Differentiating $\ln(x + \sqrt{1+x^2})$ : $\frac{1 + x/\sqrt{1+x^2}}{x + \sqrt{1+x^2}} = \frac{(\sqrt{1+x^2}+x)/\sqrt{1+x^2}}{x+\sqrt{1+x^2}} = \frac{1}{\sqrt{1+x^2}}$ . Correct.

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

Example— 5· Reduction formula and application (challenge)

Problem. Derive the reduction formula $\int \sin^n x\, dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2}x\, dx$ and use it to compute $\int_0^{\pi/2}\sin^4 x\, dx$ .

Strategy. Derive by parts; apply for $n = 4$ and $n = 2$ .

Solution.

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

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

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

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

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

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

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

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

Check. Wallis formula: $\int_0^{\pi/2}\sin^4 x\, dx = \frac{3\cdot 1}{4\cdot 2}\cdot\frac{\pi}{2} = \frac{3\pi}{16}$ . Confirms.

Source. 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
Compute $\int \sin^2 x\, dx$ .
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Ex. 87.2Application
Compute $\int \cos^2 x\, dx$ .
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Ex. 87.3ApplicationAnswer key
Compute $\int \sin^3 x\, dx$ .
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Ex. 87.4Application
Compute $\int \cos^3 x\, dx$ .
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Ex. 87.5ApplicationAnswer key
Compute $\int \sin^4 x\, dx$ .
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Ex. 87.6Application
Compute $\int \cos^4 x\, dx$ .
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Ex. 87.7Application
Compute $\int \sin^5 x\, dx$ .
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Ex. 87.8Application
Compute $\int \tan^2 x\, dx$ .
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Ex. 87.9ApplicationAnswer key
Compute $\int \sin^2 x \cos^2 x\, dx$ .
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Ex. 87.10ApplicationAnswer key
Compute $\int \sin^3 x \cos x\, dx$ .
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Ex. 87.11ApplicationAnswer key
Compute $\int \sin x \cos^3 x\, dx$ .
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Ex. 87.12Application
Compute $\int \sin^3 x \cos^2 x\, dx$ .
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Ex. 87.13Application
Compute $\int \sin(3x)\cos(2x)\, dx$ .
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Ex. 87.14Application
Compute $\int \tan^2 x \sec^2 x\, dx$ .
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Ex. 87.15Application
Compute $\int \sqrt{1 - x^2}\, dx$ .
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Ex. 87.16Application
Compute $\int \sqrt{4 - x^2}\, dx$ .
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Ex. 87.17Application
Compute $\int \dfrac{1}{\sqrt{1 - x^2}}\, dx$ .
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Ex. 87.18ApplicationAnswer key
Compute $\int \dfrac{1}{\sqrt{9 - x^2}}\, dx$ .
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Ex. 87.19Application
Compute $\int \dfrac{x^2}{\sqrt{1 - x^2}}\, dx$ .
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Ex. 87.20Application
Compute $\int \dfrac{1}{1 + x^2}\, dx$ via substitution $x = \tan\theta$ .
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Ex. 87.21Application
Compute $\int \dfrac{1}{\sqrt{1 + x^2}}\, dx$ .
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Ex. 87.22Application
Compute $\int \sqrt{1 + x^2}\, dx$ .
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Ex. 87.23ApplicationAnswer key
Compute $\int \dfrac{1}{\sqrt{x^2 - 1}}\, dx$ .
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Ex. 87.24Application
Compute $\int \sqrt{x^2 - 4}\, dx$ .
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Ex. 87.25Modeling
Prove that the area of a circle with radius $r$ is $\pi r^2$ by computing $A = 4\int_0^r \sqrt{r^2 - x^2}\, dx$ .
Ex. 87.26Modeling
Cauchy distribution: verify that $f(x) = \dfrac{1}{\pi(1+x^2)}$ satisfies $\int_{-\infty}^\infty f(x)\, dx = 1$ .
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Ex. 87.27Modeling
RMS (effective voltage) of $v(t) = V_0\sin(\omega t)$ : compute $V_{rms} = \sqrt{\frac{1}{T}\int_0^T v^2\, dt}$ where $T = 2\pi/\omega$ .
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Ex. 87.28Modeling
Area of ellipse $x^2/a^2 + y^2/b^2 = 1$ : compute $A = 4\int_0^a \frac{b}{a}\sqrt{a^2 - x^2}\, dx$ and show that $A = \pi ab$ .
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Ex. 87.29ChallengeAnswer key
Compute $\int \sec^5 x\, dx$ using the reduction formula for $\sec^n$ .
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Ex. 87.30Challenge
Compute $\int \dfrac{1}{\sin x}\, dx = \int \csc x\, dx$ using the Weierstrass substitution $t = \tan(x/2)$ .
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Ex. 87.31Proof
Proof. Prove the reduction formula $\int \sin^n x\, dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2}x\, dx$ using integration by parts.
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Ex. 87.32Proof
Proof. Show that $\int \sec x\, dx = \ln\lvert \sec x + \tan x\rvert + C$ by multiplying by $(\sec x + \tan x)/(\sec x + \tan x)$ .
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Sources

APEX Calculus v5 — Hartman et al. · 2024 · CC-BY-NC · §6.3–6.4. Primary source.
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.

Identities, patterns, and substitutions

Fundamental identities

Patterns for ∫sin⁡nxcos⁡mx dx\int \sin^n x \cos^m x\, dx∫sinnxcosmxdx

Trigonometric substitution

Reduction formulas

Worked examples

Sources

Patterns for $\int \sin^n x \cos^m x\, dx$