Created by Josh Carlson
Easy and Fun Trig Integrals
Trig Formulas to Know:
Set 2
[
Set 1
]
[
Cos2(x) =
]
[
Sin2(x) =
]
Set 3
Others You Should Already Know
1 + Tan2(x) = Sec2(x)
A: Sin2(x) + Cos2(x) = 1
1 + Cot2(x) = Csc2(x)
B: Cos2(x) – Sin2(x) = Cos(2x)
Now for the Integration
Type 1: ∫Sinn(x)dx or ∫Cosn(x)dx
If n is odd
If n is even

Take out one of the function.

Apply appropriate formula from Set 1.

Apply Formula A to the now even

Keep simplifying and applying the
powered part.

Use u-substitution to integrate.
formulas to all trig with even powers.

Integrate each term separately.
Type 2: ∫Sinm(x)Cosn(x)dx
If one is odd and >0, and the other is anything

Take out one of the odd power.

Apply Formula A to the now even part
of the one you took from.

If m and n are even

Apply appropriate formulas from Set 1
to both trig parts.

Keep simplifying and applying the
formulas to all trig with even powers.
Use u-substitution to integrate.

Integrate each term separately.
 Note: some terms may look like type 1
where n is odd, but you can do them
now!!
Created by Josh Carlson
Type 3: ∫Sin(Mx)Cos(Nx)dx or ∫Sin(Mx)Sin(Nx)dx or ∫Cos(Mx)Cos(Nx)dx

Apply the appropriate formula from Set 2, and then integrate.
Type 4: ∫Tann(x)dx or ∫Cotn(x)dx

∫Tann(x)dx → Factor out a Tan2(x) = Sec2(x) – 1, distribute, then integrate each term
separately. [Note: if u = Tan(x), then du = Sec2(x)dx]

∫Cotn(x)dx → Factor out a Cot 2(x) = Csc2(x) – 1, distribute, then integrate each term
separately. [Note: if u = Cot(x), then du = −Csc2(x)dx]
Type 5: ∫Tanm(x)Secn(x)dx or ∫Cotm(x)Cscn(x)dx
If n is even and m is any number
∫Tanm(x)Secn(x)dx

∫Cotm(x)Cscn(x)dx
Pull out one Sec2(x), and put everything

else in terms of Tan(x).

Pull out one –Csc2(x), and put
everything else in terms of Cot(x).

Distribute; integrate with u = Tan(x)
2
Distribute; integrate with u = Cot(x)
and du = –Csc2(x)dx.
and du = Sec (x)dx.
If m is odd and n is any number
∫Tanm(x)Secn(x)dx

∫Cotm(x)Cscn(x)dx
Pull out one Sec(x) and a Tan(x), group

them together, and put everything else
everything else in terms of Cot(x).

in terms of Sec(x).

Pull out one [−Cot(x)Csc(x)], and put
Distribute; integrate with u = Cot(x)
and du = −Cot(x)Csc(x)dx.
Distribute; integrate with u = Sec(x)
and du = Sec(x)Tan(x).
Rationalizing Substitutions
Anytime you see this in the integrand
Substitute this, then integrate
√
→
,
√
→
,
√
→
,