Integration of Trigonometric functions involves basic simplification techniques. These techniques use different trigonometric identities which can be written in an alternative form that are more amenable to integration.

Integration of Trigonometric Functions - Formulas, Solved Examples |

## Representation

The integration of a function f(x) is given by F(x) and it is represented by:

∫f(x)dx = F(x) + C |

Here,

R.H.S. of the equation means integral f(x) with respect to x.

F(x) is called anti-derivative or primitive.

f(x) is called the integrand.

dx is called the integrating agent.

C is called constant of integration or arbitrary constant.

x is the variable of integration.

Also, check integral formulas here.

### Integration of Trigonometric Functions Formulas

Below are the list of few formulas for the integration of trigonometric functions:

- ∫sin x dx = -cos x + C
- ∫cos x dx = sin x + C
- ∫tan x dx = ln|sec x| + C
- ∫sec x dx = ln|tan x + sec x| + C
- ∫cosec x dx = ln|cosec x – cot x| + C = ln|tan(x/2)| + C
- ∫cot x dx = ln|sin x| + C
- ∫sec
^{2}x dx = tan x + C - ∫cosec
^{2}x dx = -cot x + C - ∫sec x tan x dx = sec x + C
- ∫cosec x cot x dx = -cosec x + C
- ∫sin kx dx = -(cos kx/k) + C
- ∫cos kx dx = (sin kx/k) + C

## Related Articles

To understand this concept let us solve some examples.

## Integration of Trigonometric Functions Examples

**Example 1:**

Question- Integrate 2cos
^{2}x with respect to x.
Form this identity Substituting the above value in the given integrand, we have According to the properties of integration, the integral of sum of two functions is equal to the sum of integrals of the given functions, i.e., Therefore equation 1 can be rewritten as: This gives us the required integration of the given function. |

**Example 2:**

Question- Integrate sin 4x cos 3x with respect to x.
Therefore, From the above equation we have: According to the properties of integration, the integral of sum of two functions is equal to the sum of integrals of the given functions, i.e., Therefore equation 2 can be rewritten as: This gives us the required integration of the given function. |

**Example 3:**

Question- Integrate sin
^{2} x. cos^{2}x.
Substituting the value in the given integrand, we have Substituting the above value in equation (i), we have |

### Video Lesson on Trigonometry

### Integration of Trigonometric Functions Questions

Try solving the following practical problems on integration of trigonometric functions.

- Find the integral of (cos x + sin x).
- Evaluate: ∫(1 – cos x)/sin
^{2}x dx - Find the integral of sin
^{2}x, i.e. ∫sin^{2}x dx.

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