Integration Techniques

∫ sin^m(x)cosⁿ(x) dx

The exponent on cosx is positive and odd.

1. Factor out cosx

2. Substitute cos²x = 1 - sin²x.

3. u = sinx

∫ sin^m(x)cosⁿ(x) dx

The exponent on sinx is positive and odd.

1. Factor out sinx

2. Substitute sin²x = 1 - cos²x

3. u = cosx

∫ sin^m(x)cosⁿ(x) dx

Both exponents are positive and even.

Substitute:

sin²x → ½ (1 - cos2x)

OR

cos²x → ½ (1 + cos2x)

∫ tan^m(x)secⁿ(x) dx

The exponent on secx is positive and even.

1. Factor out sec²x

2. Substitute sec²x = tan²x + 1

3. u = tanx

∫ tan^m(x)secⁿ(x) dx

Both exponents are positive and odd.

Factor out secxtanx

1. Substitute tan²x = sec²x - 1

2. u = secx

1. Substitute x = asinθ, -π/2 ≤ θ ≤ π/2

2. Factor out a²

3. Substitute 1- sin²θ = cos²θ

1. Substitute x = atanθ, -π/2 < θ < π/2

2. Factor out a²

3. Substitute 1 + tan²θ = sec²θ

1. Substitute x = asecθ, θ ∈ [0,π/2) U [π,3π/2)

2. Factor out a²

3. Substitute sec²θ - 1 = tan²θ

∫ a^x dx

∫ tanx dx

ln|secx|

∫ cotx dx

ln|sinx|

∫ secx dx

ln|secx + tanx|

∫ cscx dx

-ln|cscx + cotx|

sin^-1x

tan^-1x

sec^-1x

d/dx sin^-1x

d/dx cos^-1x

d/dx tan^-1x

d/dx cot^-1x

d/dx sec^-1x

d/dx csc^-1x

Integration by Parts formula

Domain/range restriction for sinθ = x / sin^-1x = θ

• interval: -π/2 ≤ θ ≤ π/2

• Unit Circle: Q4 and Q1

Domain/range restriction for tanθ = x / tan^-1x = θ

• interval: -π/2 < θ < π/2

• Unit Circle: Q4 and Q1

Domain/range restriction for cosθ = x / cos^-1x = θ

• interval: 0 ≤ θ ≤ π

• Unit Circle: Q1 and Q2

Domain/range restriction for secθ = x / sec^-1x = θ

• interval: θ ∈ [0,π/2) U [π,3π/2)

• Unite Circle: Q1 and Q3

Double angle formulas

sin(2x) = 2sinxcosx

cos(2x) = cos²x - sin²x

1. numerator degree < denominator degree

2. The denominator has distinct linear factors:

   (a₁x + b₁) (a₂x + b₂) … (a_nx + b_n)

First, factor the denominator to obtain its linear factors.

numerator degree ≥ denominator degree

Use long division with the integrand.

The denominator has a repeated linear factor:

(ax + b)ⁿ

Complete the square formula for x² + bx + c

How to convert a function into a rational function using u-substitution

1. Let u equal the square root:

   ex. u = √x

2. Find u² to cancel out the radical:

   u² = (√x)²

   u² = x

3. Derive u²:

   2u du = 1dx

4. Substitute.

The denominator has a distinct irreducible factor:

ax² + bx + c

The denominator has a repeated irreducible factor:

(ax² + bx + c)ⁿ

What is the integral, with respect to x, for the area between two curves?

The functions are in terms of x:

y = x…

What is the integral, with respect to y, for the area between two curves?

The functions are in terms of y:

x = y…