Integration Formulas

∫ k dx = ?

kx + C

where k is a constant

∫ xⁿ dx = ?
Where (n ≠ −1)

[x⁽ⁿ⁺¹⁾ / (n+1)] + C.

∫ (1/x) dx = ?

ln|x| + C.

∫ e^x dx = ?

e^x + C.

∫ a^x dx = ?
Where (a > 0, a ≠ 1)

[a^x / ln(a)] + C.

∫ sin(x) dx = ?

cos(x) + C.

∫ cos(x) dx = ?

-sin(x) + C.

∫ sec²(x) dx = ?

tan(x) + C.

∫ csc²(x) dx = ?

−cot(x) + C.

∫ sec(x) tan(x) dx = ?

sec(x) + C.

∫ csc(x) cot(x) dx = ?

−csc(x) + C.

∫ tan(x) dx = ?

ln|sec(x)| + C.

∫ cot(x) dx = ?

ln|sin(x)| + C.

∫ sec(x) dx = ?

ln|sec(x) + tan(x)| + C.

∫ csc(x) dx = ?

−ln|csc(x) + cot(x)| + C.

∫ sinh(x) dx = ?

cosh(x) + C.

∫ cosh(x) dx = ?

sinh(x) + C.

∫ [1 / √(a² − x²)] dx = ?

arcsin(x/a) + C

Where (a > 0)

∫ (1 / (a² + x²)) dx = ?

(1/a) arctan(x/a) + C

Where (a > 0)

∫ [1 / (x√{x² − a²})] dx = ?

(1/a) arcsec|x/a| + C

Where (a > 0)

∫ [1 / √(a² + x²)] dx

(1/a) arcsinh(x/a) + C

Where (a > 0)

∫ [1 / √(x² − a²)] dx

arccosh(x/a) + C

Where (x > a > 0)

Integration by Parts:

∫ udv = ?

uv - ∫vdu

Fundamental Trig Identities:

cos²x + sin²x = ?

1

Fundamental Trig Identities:

1 + tan²x = ?

sec²x

Fundamental Trig Identities:

1 + cot²x = ?

csc²x

Fundamental Trig Identities:

cos²x = ?

(1/2) * (1 + cos[2x])

Fundamental Trig Identites:

sin²x = ?

(1/2) * (1 - cos[2x])

Trig Substitution:

a² - x²

x = a sinθ

dx = a cosθ dθ

Trig Substitution:

a² + x²

x = a tanθ

dx = a sec²θ dθ

Trig Substitution:

x² - a²

x = a secθ

dx = a secθtanθ dθ

Partial Fraction Decomposition Criteria (3)

1. rational function (polynomial / polynomial)

2. smaller degree in the numerator

3. denominator must factor

Partial Fraction Cases

All depend on the denominator

1. Unique Linear Factors

   (x² + 2x - 3) → (a / (x + 3)) + (b / (x - 1))

2. Nonlinear Factors

   (x - 3)(x² + 4) → (a / (x - 3)) + ((bx + c) / (x² + 4))

3. Repeating Factors

   (x - 3)³ → (a / (x - 3)) + (b / (x - 3)²) + (c / (x - 3)³)