Integral Cases for Trigonometric Powers Cheat Sheet

A Education › Mathematics cheat sheet in English by CROSSANT

Algebraic and trigonometric steps to evaluate integrals involving powers of different trigonometric functions. For problems involving powers of trigonometric functions that are both less than 4, other methods may be simpler. For large powers, reduction formulas are recommended.

Trigonometry Integrals Powers Identities Substitution Reducing Usub Pythagorean

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Lone Cases

Case Name	Step 1 - Exponent Manipulation	Step 2 - Trigonometric Identities	Step 3	Step 4
Sine-Odd	Manipulate to sin(x)(sin²(x))^p	Sine Pythagorean Id.	Let u=cos(x)
Sine-Even	Manipulate to (sin²(x))^p	Sine Power-Reducing Id.	Expand and apply integral linearity	Solve cosine cases
Cosine-Odd	Manipulate to cos(x)(cos²(x))^p	Cosine Pythagorean Id.	Let u=sin(x)
Cosine-Even	Manipulate to (cos²(x))^p	Cosine Power-Reducing Id.	Expand and apply integral linearity	Solve cosine cases
Tangent-Odd	Manipulate to tan(x)(tan²(x))^p	Tangent Pythagorean Id.	Expand and apply integral linearity	Solve other cases
Tangent-Even	Manipulate to (tan²(x))^p	Tangent Pythagorean Id.	Expand and apply integral linearity	Solve secant cases
Cosecant-Odd	Manipulate to csc(x)(csc²(x))^p	Cosecant Pythagorean Id.	Expand and apply integral linearity	Solve other cases
Cosecant-Even	Manipulate to csc²(x)(csc²(x))^p	Cosecant Pythagorean Id.	Let u=cot(x)
Secant-Odd	Manipulate to sec(x)(sec²(x))^p	Secant Pythagorean Id.	Expand and apply integral linearity	Solve other cases
Secant-Even	Manipulate to sec²(x)(sec²(x))^p	Secant Pythagorean Id.	Let u=tan(x)
Cotangent-Odd	Manipulate to cot(x)(cot²(x))^p	Cotangent Pythagorean Id.	Expand and apply integral linearity	Solve other cases
Cotangent-Even	Manipulate to (cot²(x))^p	Cotangent Pythagorean Id.	Expand and apply integral linearity	Solve cosecant cases

p ∈ ℕ₁ = {1,2,3,4,5,...}

Pair Cases

Case Name	Step 1 - Exponent Manipulation	Step 2	Step 3	Step 4
Sine-Odd and Cosine-Even	Manipulate sine powers to sin(x)(sin²(x))^p	Sine Pythagorean Id.	Let u=cos(x)
Sine-Even and Cosine-Odd	Manipulate cosine powers to cos(x)(cos²(x))^p	Cosine Pythagorean Id.	Let u=sin(x)
Sine-Odd and Cosine-Odd	Either two previous cases work
Sine-Even and Cosine-Even	Manipulate sine powers to (sin²(x))^p	Sine Pythagorean Id.	Expand and apply integral linearity	Solve cosine cases
Sine-Cosine Same	Manipulate powers to (sin(x)cos(x))^p	Sine Double-Angle Id.	Solve sine cases
Secant-Odd and Tangent-Even	Manipulate tangent powers to (tan²(x))^p	Tangent Pythagorean Id.	Expand and apply integral linearity	Solve secant cases
Secant-Even and Tangent-Odd	Manipulate powers to sec(x)tan(x)sec^p(x)tan^q(x)	Manipulate tangent powers to (tan²(x))^r	Tangent Pythagorean Id.	Let u=sec(x)
Secant-Odd and Tangent-Odd	Same as previous case
Secant-Even and Tangent-Even	Manipulate secant powers to sec²(x)(sec²(x))^p	Secant Pythagorean Id.	Let u=tan(x)
Secant-Tangent Same	Same as previous two cases
Cosecant-Odd and Cotangent-Even	Manipulate cotangent powers to (cot²(x))^p	Cotangent Pythagorean Id.	Expand and apply integral linearity	Solve cosecant cases
Cosecant-Even and Cotangent-Odd	Manipulate powers to csc(x)cot(x)csc^p(x)cot^q(x)	Manipulate cotangent powers to (cot²(x))^r	Cotangent Pythagorean Id.	Let u=csc(x)
Cosecant-Odd and Cotangent-Odd	Same as previous case
Cosecant-Even and Cotangent-Even	Manipulate cosecant powers to csc²(x)(csc²(x))^p	Cosecant Pythagorean Id.	Let u=cot(x)
Cosecant-Cotangent Same	Same as previous two cases

p, q, r ∈ ℕ₁ = {1,2,3,4,5,...}

Pythagorean Identities

Sine Pythagorean	sin²(x)=1-cos²(x)
Cosine Pythagorean	cos²(x)=1-sin²(x)
Tangent Pythagorean	tan²(x)=sec²(x)-1
Cosecant Pythagorean	csc²(x)=cot²(x)+1
Secant Pythagorean	sec²(x)=tan²(x)+1
Cotangent Pythagorean	cot²(x)=csc²(x)-1
Sine-Cosine Pythagorean	1=sin²(x)+cos²(x)
Secant-Tangent Pythagorean	1=sec²(x)-tan²(x)
Cosecant-Cotangent Pythagorean	1=csc²(x)-cot²(x)

Reduction Formulas

∫sinⁿ(x)dx, n≥2	-cos(x)sin^n-1(x)/n + (n-1/n)∫sin^n-2(x)dx
∫cosⁿ(x)dx, n≥2	cos(x)^n-1sin(x)/n + (n-1/n)∫cos^n-2(x)dx
∫tanⁿ(x)dx, n≥2	tan^n-1(x)/(n-1) - ∫tan^n-2(x)dx
∫cscⁿ(x)dx, n≥2	-csc^n-2(x)cot(x)/(n-1) + ((n-2)/(n-1))∫csc^n-2(x)dx
∫secⁿ(x)dx, n≥2	sec^n-2(x)tan(x)/(n-1) + ((n-2)/(n-1))∫sec^n-2(x)dx
∫cotⁿ(x)dx, n≥2	-cot^n-1(x)/(n-1) - ∫cot^n-2(x)dx

n ∈ ℕ₁ = {1,2,3,4,5,...}

Miscellaneous Information

Binomial Expansion	(a+b)ⁿ = ∑ⁿ_k=0 (ⁿ_k) a^n-kb^k
Integral Linearity Property: Sum and Difference	∫(f(x) ± g(x))dx = ∫f(x)dx ± ∫g(x)dx
∫tan(x)dx	ln\|sec(x)\|+C
∫cot(x)dx	ln\|sin(x)\|+C
∫sec(x)dx	ln\|sec(x)+tan(x)\|+C
∫sec³(x)dx	½(sec(x)tan(x)+ln\|sec(x)+tan(x)\|)+C
∫csc(x)dx	ln\|csc(x)-cot(x)\|+C
∫csc³(x)dx	-½(csc(x)cot(x)+ln\|csc(x)+cot(x)\|)+C

(ⁿ_k) are the binomial coefficients, equal to n!/(k!(n-k)!)
n, k ∈ ℕ₁ = {1,2,3,4,5,...}

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