Calculus II: Integration of trigonometric functions

1. Of the form: ∫ sin^o(x) cos^r (x) dx or
∫ sin^r(x) cos^o (x) dx

o stands for odd integer, and r stands for real.

The technique is:

1. Write sin^o(x) = sin^e(x) . sin(x)
(e stands for even).
2. Using the equality cos² + sin² = 1
we express sin^e(x) in function of cos (x)
sin^e(x) = [1 - cos²(x)]^e/2

Therefore

∫ sin^o(x) cos^r (x) dx =
∫ sin^e (x) sin(x) cos^r(x) dx =
∫ [1 - cos²(x)]^e/2 cos^r(x) sin(x) dx

∫ sin^o(x) cos^r (x) dx =
∫ [1 - cos²(x)]^e/2 cos^r(x) sin(x) dx

Example 1

∫ sin³(x) cos² (x) dx =
∫ [1 - cos²(x)]^2/2 cos²(x) sin(x) dx =
∫ [cos²(x) - cos⁴(x)] sin(x) dx =
∫cos²(x) sin(x) dx - ∫cos⁴(x) sin(x) dx

Let
cos(x) = u
so du = - sin(x) dx ,

we obtain:

- ∫u² du + ∫u⁴du = - (1/3)u³ + (1/5)u⁵ =

- (1/3) cos³(x) + (1/5) cos⁵(x) + cst

∫ sin³(x) cos² (x) dx =
- (1/3) cos³(x) + (1/5) cos⁵(x) + cst

2. Of the form: ∫ sin^e(x) cos^e (x) dx or
∫ sin^e(x) dx or ∫ cos^e (x) dx

e stands for even

In these kinds, we use the three following identities:

1. cos²(x) = (1 + cos(2x))/2
2. sin²(x) = (1 - cos(2x))/2
3. sin(2x) = 2 sin(x) cos(x)

Example

I = ∫ sin⁴(x) cos²(x) dx

We have:

sin⁴(x) = (1 - cos(2x))²/4
cos²(x) = (1 + cos(2x))/2

Therefore

sin⁴(x) cos²(x) =
(1/8) (1 - cos(2x))² .(1 + cos(2x)) =
(1/8) (1 - cos(2x)) . (1 - cos(2x)) (1 + cos(2x)) =
(1/8) (1 - cos(2x)) . (1 - cos²(2x)) =
(1/8) [1 - cos²(2x) - cos(2x) + cos³(2x)] =
(1/8) [cos³(2x) - cos²(2x) - cos(2x) + 1]

Therefore

I = (1/8) [∫ cos³(2x) dx - ∫ cos²(2x) dx - ∫ cos(2x) dx + ∫1 dx]

1. First term:

cos³(2x) = cos²(2x) cos(2x) =
(1 - sin²(2x)) cos(2x) =
cos(2x) - sin²(2x)) cos(2x)
∫ cos³(2x) dx =
∫ cos(2x) dx - ∫ sin²(2x)) cos(2x) dx =
(1/2)∫ cos(2x) d(2x) - (1/2)∫ sin²(2x)) dsin(2x) =
(1/2) sin(2x) - (1/2) (1/3) sin³(2x)).

∫ cos³(2x) dx = cos²(2x) cos(2x) = (1/2)∫ cos(2x) d(2x) - (1/2) (1/3) sin³(2x))

2. Second term:

cos²(2x) = (1 + cos(4x))/2

(1/2)∫ (1 + cos(4x)) dx = (1/2) [∫ (1 + cos(4x))] dx =
(1/2) [∫dx + ∫cos(4x))dx = (1/2) [x + (1/4)sin(4x)]

(1/2)∫ (1 + cos(4x)) dx = (1/2) [x + (1/4)sin(4x)]

3. Third term::

∫ cos(2x) dx = (1/2) sin(2x)

4 Fourth term::

∫1 dx = x

Therefore

I = (1/8) [(1/2)sin(2x) - (1/2) (1/3) sin³(2x)) - (1/2) (x + (1/4)sin(4x)) - (1/2) sin(2x) + x ] + cst.
= (1/16) [sin(2x) - (1/3) sin³(2x)) - x - (1/4)sin(4x) - sin(2x) + 2x ] + cst =
(1/16) [ - (1/3) sin³(2x)) - (1/4)sin(4x) + x] + cst

∫ sin⁴(x) cos²(x) dx =
(1/16) [ - (1/3) sin³(2x)) - (1/4)sin(4x) + x] + cst

3. Of the form: ∫ tan^r(x) sec^e (x) dx or
∫ cotg^r(x) cosec^e (x) dx

r stands for real, and e stands for even

The steps are:

1. We start to transform sec^e (x) into sec^{e - 2} (x) . sec² (x)

2. Using sec²(x) = 1 + tan²(x), we express sec^{e - 2} (x) in a function of tan(x)

3. For the second kind of integral, we use cosec²(x) = 1 + cotg²(x) instead.

Example

∫ tan^r(x) sec⁴(x) dx =

∫ tan^r(x) sec²(x) sec²(x) dx =
∫ tan^r(x) [1 + tan²(x)] sec²(x) dx =
∫ tan^r(x) [1 + tan²(x)] sec²(x) dx =
= ∫ tan^r(x) sec²(x) dx + ∫ tan^r+2(x) sec²(x) dx

Let u = tan(x), so du = sec²(x) dx
= ∫ u^r du + ∫ u^r+2 du =
(1/(r + 1))u^r+1 + (1/(r + 2))u^r+2 =
(1/(r + 1)) tan^r+1(x) + (1/(r + 2)) tan^r+2(x) + cst.

∫ tan^r(x) sec⁴(x) dx = (1/(r + 1)) tan^r+1(x) + (1/(r + 2)) tan^r+2(x) + cst

4. Of the form: ∫ tan^o(x) sec^r(x) dx or
∫ cotg^o(x) cosec^r (x) dx

r stands for real, and o stands for odd

The steps are:

1. We start to transform tan^o(x) sec^r(x) into tan^o-1(x) sec^r-1(x) . tan(x) sec(x)

2. Using sec²(x) = 1 + tan²(x), we express then tan^{o - 1}(x) in a function of sec(x)

3. For the second kind of integral, we use cosec²(x) = 1 + cotg²(x) instead.

Example

∫ tan³(x) sec^r(x) dx =
∫ tan²(x) sec^r-1(x) tan(x) sec(x) dx =
∫ [sec²(x) - 1] sec^r-1(x) tan(x) sec(x) dx =
=
∫ sec²(x) . sec^r-1(x) tan(x) sec(x) dx -
∫ sec^r-1(x) . tan(x) sec(x) dx =
∫ sec^r+1(x) tan(x) sec(x) dx
- ∫ sec^r-1(x) . tan(x) sec(x) dx =

Let u = sec(x), so du = sec(x) tan(x) dx

= ∫ u^r+1 du - ∫ u^r-1 du =
(1/(r+2))u^r+2 - (1/(r)) u^r = (1/(r+2)) sec^r+2(x) - (1/(r)) sec^r(x) + cst.

∫ tan³(x) sec^r(x) dx = (1/(r+2)) sec^r+2(x) - (1/(r)) sec^r(x) + cst.