Integration involving Trigonometric Functions
June 16, 2025
Introduction
When you start integrating trigonometric functions, the key is to recognize their standard forms - like \text{sin }(x),\text{ cos }(x), \text{ and tan }(x),. You'll see these pop up often, especially integrals that require more complex techniques. The good news is, once we get familiar with how they behave, integrating them becomes straightforward.
Here are some of the standard integral formulas involving trigonometric functions:
- \displaystyle \int \cos(u)\,du = -\cos(u) + C
- \displaystyle \int csc(u) \space \cot(u)\,du = -csc(u) + C
- \displaystyle \int sin(u) \,du = -cos(u) + C
- \displaystyle \int \sec(u) \space tan(u)\,du = sec(u) + C
- \displaystyle \int \sec^2(u)\,du = \tan(u) +C
- \displaystyle \int \csc^2(u)\,du = -cot(u) + C
- \displaystyle \int \tan(u)\,du = -\ln|\csc (u)| +C
- \displaystyle \int \tan(u)\,du = \ln|\sin(u)| + C
- \displaystyle \int \cot(u)\,du = \ln|\csc(u) + C
- \displaystyle \int \cot(u)\,du = \ln|\sin(u)| + C
- \displaystyle \int \sec(u)\,du = \ln|\sec(u) + \tan(u)| + C
- \displaystyle \int \csc(u)\,du =\ln|\csc(u) - |cot(u)| + C
- \displaystyle \int \sin^2(u)\,du = \dfrac{u}{2} - \dfrac{\sin(2u)}{4} + C
- \displaystyle \int \cos^2(u)\,du = \dfrac{u}{2} + \dfrac{\sin(2u)}{4} + C
The illustrative examples below will give you a detailed way to evaluate such integrals:
Illustrative Example #1
Evaluate the integral \displaystyle \int (2 \sin x + 3 \cos x)\,dx.
This integral is a direct sum of standard trigonometric functions. We can integrate each term separately using basic integration rules:
Now, apply the standard integrals:
So we get:
Therefore,
Basic integration rules still apply! You can use the sum/difference rule,
and the constant multiple rule,
when integrating expressions involving trigonometric, polynomial, or exponential functions.
Illustrative Example #2
Evaluate the integral \displaystyle \int (4 \cos x - \sin x)\, dx.
This integral involves a sum and difference of standard trigonometric functions. We apply the sum/difference rule and constant multiple rule:
Now use the basic integration formulas:
So we get:
Therefore,
Illustrative Example #3
Evaluate the integral \displaystyle \int (3 \sec^2 x + 2)\,dx.
This integral contains a sum of functions, one of which is a standard trigonometric derivate. Apply the sum and constant multiple rules:
Use the standard integrals:
So we have:
Therefore:
Illustrative Example #4
Evaluate the integral \displaystyle \int (4 + \tan x) \, dx.
This is a sum of a constant and a standard trigonometric function. Use the sum and constant multiple rules:
Apply the basic integrals:
So we get:
Therefore,
Illustrative Example #5
Evaluate the integral \displaystyle \int (3\tan x + 3\sec^2 x) \, dx.
This integral involves a sum of two standard trigonometric functions. Factor out the common constant and apply the sum rule:
Use the standard integration formulas:
So we have:
Therefore,
Illustrative Example #6
Evaluate the integral \displaystyle \int (2\csc x + 3\cot x) \, dx .
This integral involves a sum of trigonometric functions. Apply the sum rule and constant multiple rule:
Use the standard integrals:
So we get:
Therefore,
Illustrative Example #7
Evaluate the integral \displaystyle \int \sec(x) + \sin(x) \, dx.
This integral involves a sum of two basic trigonometric functions. Use the sum rule:
Recall the standard integrals:
So we have:
Therefore,
Illustrative Example #8
Evaluate the integral \displaystyle \int 5\csc(x) - 3\cos(x) + \pi \tan(x) \, dx.
This integral involves multiple trigonometric terms with constants. Apply the sum/difference and constant multiple rules:
Recall the standard integrals:
Now substitute each result:
Therefore,
Illustrative Example #9
Evaluate the integral \displaystyle \int 12\sin^2(x) \, dx.
Begin by applying the constant multiple rule:
Now use the standard integral formula:
Substitute this into the expression:
Simplify:
Therefore,
