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Home/Calculus/Indefinite Integration/Special Integrals

CALC-5.5

5-6 hours

Special Integrals

Advanced techniques: partial fractions, trigonometric integrals, and reduction formulas.

Learning Objectives

Decompose rational functions using partial fractions
Integrate products of sines and cosines
Apply Weierstrass substitution for rational trig functions
Use reduction formulas for powers of trig functions
Handle integrals with quadratic expressions

1. Partial Fractions

Theorem 5.16: Partial Fraction Decomposition

For a proper rational function P(x)/Q(x) where Q(x) factors as:

Distinct linear: $(x-a)(x-b) \to \frac{A}{x-a} + \frac{B}{x-b}$
Repeated linear: $(x-a)^2 \to \frac{A}{x-a} + \frac{B}{(x-a)^2}$
Irreducible quadratic: $(x^2+bx+c) \to \frac{Ax+B}{x^2+bx+c}$

Example 5.50: Distinct Linear Factors

Problem: Evaluate $\int \frac{1}{x^2-1}\,dx$

Solution:

Factor: $x^2-1 = (x-1)(x+1)$

\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}

Solving: A = 1/2, B = -1/2

\int \frac{1}{x^2-1}\,dx = \frac{1}{2}\ln|x-1| - \frac{1}{2}\ln|x+1| + C = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C

Example 5.51: Repeated Linear Factor

Problem: Evaluate $\int \frac{x+3}{(x+1)^2}\,dx$

Solution:

\frac{x+3}{(x+1)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2}

Solving: A = 1, B = 2

\int \left(\frac{1}{x+1} + \frac{2}{(x+1)^2}\right)\,dx = \ln|x+1| - \frac{2}{x+1} + C

Example 5.52: Irreducible Quadratic

Problem: Evaluate $\int \frac{1}{x^2+4}\,dx$

Solution:

The quadratic is irreducible. Use arctan formula:

\int \frac{1}{x^2+4}\,dx = \frac{1}{2}\arctan\frac{x}{2} + C

2. Trigonometric Integrals

Strategy for ∫sin^n x cos^m x dx

Case 1: n odd

Save one sin x, convert rest to cos using sin²x = 1 - cos²x, substitute u = cos x

Case 2: m odd

Save one cos x, convert rest to sin using cos²x = 1 - sin²x, substitute u = sin x

Case 3: Both even

Use half-angle: sin²x = (1-cos 2x)/2, cos²x = (1+cos 2x)/2

Example 5.53: sin³x dx

Problem: Evaluate $\int \sin^3 x\,dx$

Solution:

n = 3 is odd. Write $\sin^3 x = \sin x \cdot \sin^2 x = \sin x(1-\cos^2 x)$

Let u = cos x, du = -sin x dx:

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

= -\cos x + \frac{\cos^3 x}{3} + C

Example 5.54: sin²x dx

Problem: Evaluate $\int \sin^2 x\,dx$

Solution:

Both powers even. Use half-angle: $\sin^2 x = \frac{1-\cos 2x}{2}$

\int \frac{1-\cos 2x}{2}\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C

Example 5.55: tan²x dx

Problem: Evaluate $\int \tan^2 x\,dx$

Solution:

Use identity: $\tan^2 x = \sec^2 x - 1$

\int (\sec^2 x - 1)\,dx = \tan x - x + C

3. Weierstrass Substitution

Theorem 5.17: Weierstrass Substitution

Let $t = \tan\frac{x}{2}$ . Then:

\sin x = \frac{2t}{1+t^2}

\cos x = \frac{1-t^2}{1+t^2}

dx = \frac{2\,dt}{1+t^2}

This converts rational functions of sin x and cos x to rational functions of t.

Example 5.56: Weierstrass Application

Problem: Evaluate $\int \frac{1}{1+\sin x}\,dx$

Solution:

Let t = tan(x/2):

\int \frac{1}{1+\frac{2t}{1+t^2}} \cdot \frac{2}{1+t^2}\,dt = \int \frac{2}{(1+t)^2}\,dt = -\frac{2}{1+t} + C

Back-substitute: $= -\frac{2}{1+\tan(x/2)} + C$

4. Reduction Formulas

Theorem 5.18: Common Reduction Formulas

For sin^n x:

\int \sin^n x\,dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int \sin^{n-2}x\,dx

For cos^n x:

\int \cos^n x\,dx = \frac{\cos^{n-1}x\sin x}{n} + \frac{n-1}{n}\int \cos^{n-2}x\,dx

For tan^n x:

\int \tan^n x\,dx = \frac{\tan^{n-1}x}{n-1} - \int \tan^{n-2}x\,dx

For sec^n x:

\int \sec^n x\,dx = \frac{\sec^{n-2}x\tan x}{n-1} + \frac{n-2}{n-1}\int \sec^{n-2}x\,dx

Example 5.57: Using Reduction Formula

Problem: Evaluate $\int \sin^4 x\,dx$

Solution:

Using the reduction formula twice:

\int \sin^4 x\,dx = -\frac{\sin^3 x \cos x}{4} + \frac{3}{4}\int \sin^2 x\,dx

= -\frac{\sin^3 x \cos x}{4} + \frac{3}{4}\left(\frac{x}{2} - \frac{\sin 2x}{4}\right) + C

= -\frac{\sin^3 x \cos x}{4} + \frac{3x}{8} - \frac{3\sin 2x}{16} + C

5. Completing the Square

Example 5.58: Completing the Square

Problem: Evaluate $\int \frac{1}{x^2+6x+13}\,dx$

Solution:

Complete the square: $x^2+6x+13 = (x+3)^2 + 4$

Let u = x + 3:

\int \frac{1}{u^2+4}\,du = \frac{1}{2}\arctan\frac{u}{2} + C = \frac{1}{2}\arctan\frac{x+3}{2} + C

Example 5.59: Splitting the Numerator

Problem: Evaluate $\int \frac{2x+5}{x^2+4x+8}\,dx$

Solution:

Note that d/dx(x²+4x+8) = 2x+4. Split: 2x+5 = (2x+4) + 1

\int \frac{2x+4}{x^2+4x+8}\,dx + \int \frac{1}{x^2+4x+8}\,dx

First integral: ln|x²+4x+8|. Second: complete square (x+2)²+4

= \ln(x^2+4x+8) + \frac{1}{2}\arctan\frac{x+2}{2} + C

6. More Partial Fraction Examples

Example 5.71: Repeated Linear Factors

Problem: $\int \frac{x+2}{(x-1)^2(x+1)}\,dx$

Decompose: $\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+1}$

Solving: A = 1/4, B = 3/2, C = -1/4

= \frac{1}{4}\ln|x-1| - \frac{3}{2(x-1)} - \frac{1}{4}\ln|x+1| + C

Example 5.72: Irreducible Quadratic

Problem: $\int \frac{2x+3}{(x^2+1)(x-2)}\,dx$

Decompose: $\frac{Ax+B}{x^2+1} + \frac{C}{x-2}$

= \frac{1}{5}\ln(x^2+1) - \frac{1}{5}\arctan x + \frac{7}{5}\ln|x-2| + C

Example 5.73: Higher Degree Numerator

Problem: $\int \frac{x^3}{x^2-1}\,dx$

First divide: x³/(x²-1) = x + x/(x²-1)

= \frac{x^2}{2} + \frac{1}{2}\ln|x^2-1| + C

7. More Trigonometric Integrals

Example 5.74: sin⁴x

Problem: $\int \sin^4 x\,dx$

Use sin²x = (1-cos2x)/2 twice:

\sin^4 x = \frac{(1-\cos 2x)^2}{4} = \frac{1-2\cos 2x + \cos^2 2x}{4}

= \frac{3x}{8} - \frac{\sin 2x}{4} + \frac{\sin 4x}{32} + C

Example 5.75: sin³x cos³x

Problem: $\int \sin^3 x \cos^3 x\,dx$

Save cos x, convert sin³x:

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

Let u = cos x:

= -\frac{\cos^4 x}{4} + \frac{\cos^6 x}{6} + C

Example 5.76: tan⁴x

Problem: $\int \tan^4 x\,dx$

Write tan⁴x = tan²x(sec²x - 1):

= \int \tan^2 x \sec^2 x\,dx - \int \tan^2 x\,dx

= \frac{\tan^3 x}{3} - \tan x + x + C

Example 5.77: sec⁴x

Problem: $\int \sec^4 x\,dx$

Write sec⁴x = sec²x(1 + tan²x):

= \int \sec^2 x\,dx + \int \tan^2 x \sec^2 x\,dx

= \tan x + \frac{\tan^3 x}{3} + C

8. Practice Problems

1. $\int \frac{3x+5}{x^2-4}\,dx$

Answer

$\frac{11}{4}\ln|x-2| + \frac{1}{4}\ln|x+2| + C$

2. $\int \sin^5 x\,dx$

Answer

$-\cos x + \frac{2\cos^3 x}{3} - \frac{\cos^5 x}{5} + C$

3. $\int \frac{1}{x^2+6x+13}\,dx$

Answer

$\frac{1}{2}\arctan\frac{x+3}{2} + C$

4. $\int \cos^4 x\,dx$

Answer

$\frac{3x}{8} + \frac{\sin 2x}{4} + \frac{\sin 4x}{32} + C$

5. $\int \frac{x^2+1}{x^3+x}\,dx$

Answer

$\ln|x| + \arctan x + C$

9. Study Tips

1. Identify Type First

Rational? Trig powers? Radicals? Choose method accordingly.

2. Factor Completely

For partial fractions, factor the denominator fully first.

3. Trig Identities

Know sin²x + cos²x = 1, half-angle formulas, etc.

4. Cover-up Method

Quick way to find coefficients for distinct linear factors.

5. Complete Square

Transforms ax²+bx+c into (x+p)²+q form.

6. Verify

Always differentiate to check your answer.

10. Formula Summary

Integrand	Method
P(x)/Q(x), deg P < deg Q	Partial fractions
sinᵐx cosⁿx (odd power)	Save one, convert rest
sinᵐx cosⁿx (both even)	Half-angle formulas
tanᵐx secⁿx	Strategic save + identity
1/(a+b sin x), etc.	Weierstrass: t = tan(x/2)
1/(ax²+bx+c)	Complete square → arctan

11. More Worked Examples

Example 5.78: cot⁴x

\int \cot^4 x\,dx = \int (\csc^2 x - 1)\cot^2 x\,dx

= -\frac{\cot^3 x}{3} - \cot x - x + C

Example 5.79: 1/(sin x + cos x)

Using Weierstrass t = tan(x/2):

\int \frac{1}{\sin x + \cos x}\,dx = \frac{1}{\sqrt{2}}\ln\left|\tan\frac{x}{2} + 1 - \sqrt{2}\right| + C

Example 5.80: Complex Partial Fractions

Problem: $\int \frac{2x^2+x+1}{(x+1)(x^2+1)}\,dx$

= \ln|x+1| + \frac{1}{2}\ln(x^2+1) + C

12. Final Practice Set

P1: $\int \frac{x}{(x+1)(x+2)}\,dx$

Answer

$-\ln|x+1| + 2\ln|x+2| + C$

P2: $\int \sin^2 x \cos^4 x\,dx$

Answer

$\frac{x}{16} - \frac{\sin 4x}{64} + \frac{\sin^3 2x}{48} + C$

P3: $\int \tan^3 x \sec x\,dx$

Answer

$\frac{\sec^3 x}{3} - \sec x + C$

P4: $\int \frac{1}{(x^2+4)^2}\,dx$

Answer

$\frac{x}{8(x^2+4)} + \frac{\arctan(x/2)}{16} + C$

13. Chapter Summary

Partial Fractions

• Linear: A/(ax+b)
• Repeated: A/(ax+b) + B/(ax+b)²
• Irreducible: (Ax+B)/(ax²+bx+c)

Trig Integrals

• Odd power: save one, use identity
• Both even: half-angle formulas
• tan/sec: strategic approach

14. More Worked Examples

Example 5.81: csc⁴x

\int \csc^4 x\,dx = -\cot x - \frac{\cot^3 x}{3} + C

Example 5.82: Repeated Irreducible Quadratic

Problem: $\int \frac{x}{(x^2+1)^2}\,dx$

Let u = x²+1:

= -\frac{1}{2(x^2+1)} + C

Example 5.83: sin²x cos⁴x

Both powers even, use half-angle:

\int \sin^2 x \cos^4 x\,dx = \frac{x}{16} - \frac{\sin 4x}{64} + \frac{\sin^3 2x}{48} + C

Example 5.84: tan⁵x sec³x

Save sec x tan x, convert tan⁴x = (sec²x-1)²:

= \frac{\sec^7 x}{7} - \frac{2\sec^5 x}{5} + \frac{\sec^3 x}{3} + C

15. Challenge Problems

C1: $\int \frac{1}{x^4+1}\,dx$

Hint

Factor x⁴+1 = (x²+√2x+1)(x²-√2x+1)

C2: $\int \frac{\sin x}{1+\sin x}\,dx$

Hint

Multiply by (1-sin x)/(1-sin x) or use Weierstrass

C3: $\int \sec^6 x\,dx$

Hint

Write sec⁶x = sec⁴x·sec²x = (1+tan²x)²sec²x

16. Quick Reference Card

Partial Fractions

• Long divide if deg(P) ≥ deg(Q)
• Factor Q(x) completely
• Cover-up for distinct linear

Trig Powers

• sinᵐx cosⁿx: save odd
• Both even: half-angle
• tanᵐx secⁿx: strategic

Special

• Weierstrass: t = tan(x/2)
• Complete square for ax²+bx+c
• Always verify!

17. Additional Practice

$\int \frac{x^2-1}{x^4+x^2+1}\,dx$

Answer

$\arctan\frac{x^2+1}{x} + C$

$\int \sin^3 x \cos^5 x\,dx$

Answer

$-\frac{\cos^6 x}{6} + \frac{\cos^8 x}{8} + C$

$\int \frac{x+1}{x^2(x-1)}\,dx$

Answer

$-\frac{1}{x} + \ln|x-1| + C$

$\int \tan^2 x \sec^4 x\,dx$

Answer

$\frac{\tan^3 x}{3} + \frac{\tan^5 x}{5} + C$

18. Important Results

\int \frac{1}{x^2+a^2}\,dx = \frac{1}{a}\arctan\frac{x}{a} + C

\int \frac{1}{x^2-a^2}\,dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C

\int \frac{1}{\sqrt{a^2-x^2}}\,dx = \arcsin\frac{x}{a} + C

\int \frac{1}{\sqrt{x^2 \pm a^2}}\,dx = \ln|x + \sqrt{x^2 \pm a^2}| + C

19. Final Practice Set

F1: $\int \frac{2x+3}{x^2+3x+2}\,dx$

Answer

$\ln|x+1| + \ln|x+2| + C$

F2: $\int \cot^3 x \csc x\,dx$

Answer

$-\frac{\csc^3 x}{3} + \csc x + C$

F3: $\int \frac{x^3+2}{x^2+x+1}\,dx$

Hint

Long divide first, then partial fractions

20. Additional Worked Examples

Example 5.85: sin⁶x

Use sin²x = (1-cos2x)/2 repeatedly:

\int \sin^6 x\,dx = \frac{5x}{16} - \frac{15\sin 2x}{64} + \frac{3\sin 4x}{64} - \frac{\sin 6x}{192} + C

Example 5.86: 1/(1+sin x+cos x)

Weierstrass: t = tan(x/2)

\int \frac{1}{1+\sin x+\cos x}\,dx = \ln|1+\tan(x/2)| + C

Example 5.87: Complex Partial Fraction

Problem: $\int \frac{x^2+1}{(x-1)^2(x+2)}\,dx$

= \frac{2}{9}\ln|x-1| - \frac{2}{3(x-1)} + \frac{5}{9}\ln|x+2| + C

21. More Practice Problems

$\int \frac{x^2}{(x+1)(x^2+4)}\,dx$

Answer

$\frac{1}{5}\ln|x+1| + \frac{2}{5}\ln(x^2+4) - \frac{1}{5}\arctan\frac{x}{2} + C$

$\int \sin^4 x \cos^2 x\,dx$

Answer

$\frac{x}{16} - \frac{\sin 4x}{64} - \frac{\sin^3 2x}{48} + C$

$\int \frac{1}{\sin x - \cos x}\,dx$

Answer

$-\frac{1}{\sqrt{2}}\ln|\csc(x+\pi/4) + \cot(x+\pi/4)| + C$

$\int \cot^5 x\,dx$

Answer

$-\frac{\cot^4 x}{4} + \frac{\cot^2 x}{2} + \ln|\sin x| + C$

22. Final Chapter Summary

Partial Fractions

• Factor denominator
• Decompose to simpler terms
• Integrate each separately

Trig Powers

• Odd: save one, convert
• Even: half-angle
• tan/sec: strategic

Special Methods

• Weierstrass for rational trig
• Complete square for quadratics
• Always verify answers

23. Even More Examples

\int \frac{1}{(x^2+1)^2}\,dx = \frac{x}{2(x^2+1)} + \frac{1}{2}\arctan x + C

\int \tan^6 x\,dx = \frac{\tan^5 x}{5} - \frac{\tan^3 x}{3} + \tan x - x + C

\int \frac{1}{x^3-1}\,dx = \frac{1}{3}\ln|x-1| - \frac{1}{6}\ln(x^2+x+1) - \frac{\sqrt{3}}{3}\arctan\frac{2x+1}{\sqrt{3}} + C

\int \sec^5 x\,dx = \frac{\sec^3 x\tan x}{4} + \frac{3\sec x\tan x}{8} + \frac{3}{8}\ln|\sec x+\tan x| + C

24. More Practice Problems

M1: $\int \frac{x^3+1}{x^3-x}\,dx$

Hint

Long divide first, factor x³-x = x(x-1)(x+1)

M2: $\int \sin^7 x\,dx$

Hint

Save sin x, convert sin⁶x = (1-cos²x)³

M3: $\int \frac{1}{x^4+4}\,dx$

Hint

Factor x⁴+4 = (x²+2x+2)(x²-2x+2)

25. Final Worked Examples

Example 5.88: cos⁶x

Use cos²x = (1+cos2x)/2 repeatedly:

\int \cos^6 x\,dx = \frac{5x}{16} + \frac{15\sin 2x}{64} + \frac{3\sin 4x}{64} + \frac{\sin 6x}{192} + C

Example 5.89: 1/(x⁴+x²+1)

Factor: x⁴+x²+1 = (x²+x+1)(x²-x+1)

\int \frac{1}{x^4+x^2+1}\,dx = \frac{1}{2\sqrt{3}}\arctan\frac{2x+1}{\sqrt{3}} + \frac{1}{2\sqrt{3}}\arctan\frac{2x-1}{\sqrt{3}} + C

26. Method Selection Guide

Integrand Type	Recommended Method
Rational P(x)/Q(x)	Partial Fractions
sinᵐx cosⁿx (odd power)	Save one, convert rest
sinᵐx cosⁿx (both even)	Half-angle formulas
tanᵐx secⁿx	Strategic save + identity
R(sin x, cos x)	Weierstrass: t = tan(x/2)

27. Final Examples

∫csc⁴x dx

= -\cot x - \frac{\cot^3 x}{3} + C

∫1/(x³+1) dx

= \frac{1}{3}\ln|x+1| - \frac{1}{6}\ln(x^2-x+1) + \frac{\sqrt{3}}{3}\arctan\frac{2x-1}{\sqrt{3}} + C

28. Additional Practice Problems

$\int \frac{x^4-1}{x^2(x^2+1)}\,dx$

Answer

$x + \frac{1}{x} - 2\arctan x + C$

$\int \tan^5 x\,dx$

Answer

$\frac{\tan^4 x}{4} - \frac{\tan^2 x}{2} - \ln|\cos x| + C$

$\int \frac{1}{1-\cos x}\,dx$

Answer

$-\cot(x/2) + C$

$\int \frac{\sin^3 x}{1+\cos^2 x}\,dx$

Hint

u = cos x, sin²x = 1-cos²x

29. Additional Results

\int \tan^n x\,dx = \frac{\tan^{n-1}x}{n-1} - \int \tan^{n-2}x\,dx

\int \cot^n x\,dx = -\frac{\cot^{n-1}x}{n-1} - \int \cot^{n-2}x\,dx

\int \csc^n x\,dx = -\frac{\csc^{n-2}x\cot x}{n-1} + \frac{n-2}{n-1}\int \csc^{n-2}x\,dx

\int \frac{1}{(x^2+a^2)^n}\,dx = \frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac{2n-3}{2(n-1)a^2}\int \frac{dx}{(x^2+a^2)^{n-1}}

30. Final Chapter Summary

Partial Fractions

• Factor denominator
• Decompose
• Integrate each part

Trig Powers

• Odd: save one
• Even: half-angle
• tan/sec: strategic

Special Methods

• Weierstrass
• Complete square
• Reduction formulas

31. Final Practice Set

$\int \frac{x^3}{x^4-1}\,dx$

Answer

$\frac{1}{4}\ln|x^4-1| + C$

$\int \sin^8 x\,dx$

Hint

Use half-angle repeatedly: sin²x = (1-cos2x)/2

$\int \frac{1}{(x-1)^3(x+2)}\,dx$

Hint

A/(x-1) + B/(x-1)² + C/(x-1)³ + D/(x+2)

$\int \tan^7 x\,dx$

Hint

tan⁷x = tan⁵x(sec²x-1)

32. Important Trig Identities

$\sin^2 x = \frac{1-\cos 2x}{2}$

$\cos^2 x = \frac{1+\cos 2x}{2}$

$\tan^2 x = \sec^2 x - 1$

$\cot^2 x = \csc^2 x - 1$

33. Weierstrass Substitution Reference

Let $t = \tan(x/2)$ , then:

\sin x = \frac{2t}{1+t^2}

\cos x = \frac{1-t^2}{1+t^2}

\tan x = \frac{2t}{1-t^2}

dx = \frac{2}{1+t^2}\,dt

34. Final Exercises

$\int \frac{1}{2+\sin x}\,dx$

Hint

Use Weierstrass: t = tan(x/2)

$\int \sin^3 x \cos^4 x\,dx$

Hint

Save sin x, convert sin²x = 1-cos²x

$\int \frac{x^2+3x-2}{(x-1)(x^2+4)}\,dx$

Hint

Partial fractions: A/(x-1) + (Bx+C)/(x²+4)

$\int \sec^3 x\,dx$

Answer

$\frac{1}{2}\sec x\tan x + \frac{1}{2}\ln|\sec x+\tan x| + C$

Special Integrals Quiz

Questions

Correct

Accuracy

\int \frac{1}{(x-1)(x+2)}\,dx

partial fractions gives:

Medium

Not attempted

\int \sin^2 x\,dx =

Easy

Not attempted

\int \sin^3 x\,dx

uses:

Medium

Not attempted

The Weierstrass substitution is:

Hard

Not attempted

\int \tan^2 x\,dx =

Medium

Not attempted

For

\int \frac{2x+3}{x^2+x+1}\,dx

Hard

Not attempted

\int \sec^3 x\,dx

requires:

Hard

Not attempted

\int \frac{1}{x^2+2x+5}\,dx

first step:

Medium

Not attempted

Frequently Asked Questions

When do I use partial fractions?

For rational functions P(x)/Q(x) where deg(P) < deg(Q) and Q can be factored. Factor Q into linear and irreducible quadratic factors.

What if degree of numerator ≥ denominator?

First do polynomial long division to get a polynomial plus a proper fraction, then apply partial fractions to the fraction.

How do I integrate sin^n x cos^m x?

If n or m is odd, save one factor and convert rest using sin²+cos²=1. If both even, use half-angle formulas.

What is the Weierstrass substitution?

t = tan(x/2) gives: sin x = 2t/(1+t²), cos x = (1-t²)/(1+t²), dx = 2dt/(1+t²). Converts any rational trig to rational function.

When do I complete the square?

When the denominator is an irreducible quadratic ax² + bx + c. Completing the square reveals the arctan or arcsin form.

What are reduction formulas?

Recurrence relations like ∫sin^n x dx = -sin^{n-1}x cos x/n + (n-1)/n ∫sin^{n-2}x dx that reduce power.

Key Takeaways

Partial Fractions

Factor denominator, decompose, integrate term by term

Trig Powers

Odd power: save one, convert rest. Both even: half-angle

Weierstrass

t = tan(x/2) for rational trig functions

Complete Square

For irreducible quadratics in denominator

Identify the integral type first
Factor denominators completely
Know your trig identities
Always verify by differentiating