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

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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.Solution- To integrate the given trigonometric functions we will use the trigonometric identity –cos2x=(1+cos2×2)Form this identity2cos2x=1+cos2xSubstituting the above value in the given integrand, we have∫2cos2xdx=∫(1+cos2x).dx…(1)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.,∫[f(x)+g(x)]dx=∫f(x).dx+∫g(x).dxTherefore equation 1 can be rewritten as:∫(1+cos2x)dx=∫1dx+∫cos2xdx=x+sin2×2+CThis gives us the required integration of the given function. |

**Example 2:**

Question- Integrate sin 4x cos 3x with respect to x.Solution- To integrate the trigonometric function we will use the trigonometric identity:sinxcosy=12[sin(x+y)+sin(x−y)]Form this identitysin4xcos3x=12(sin7x+sinx)Therefore,∫(sin4xcos3x)dx=∫12(sin7x+sinx)dxFrom the above equation we have:∫12(sin7x+sinx)dx=12∫(sin7x+sinx)dx…(ii)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.,∫[f(x)+g(x)]dx=∫f(x).dx+∫g(x).dxTherefore equation 2 can be rewritten as:12∫(sin7x)+12∫(sinx)dx=−cos7×14+−cosx2+CThis gives us the required integration of the given function. |

**Example 3:**

Question- Integrate sin^{2} x. cos^{2}x.Solution- Before integration let us use few trigonometric relations in order to simplify the integrand.We know, 2sinxcosx=sin2xsinx.cosx=sin2x2Substituting the value in the given integrand, we have∫sin2x.cos2xdx=∫(sinx.cosx)2dx=∫(sin2×2)2=14∫sin22x…(iAlso we know, sin2x=1–cos2x2Substituting the above value in equation (i), we have14∫sin22x=14∫1−cos4×2=∫18dx–∫cos4x8dx=18x+C1–sin4×32+C2=18x–sin4×32+C |

### Video Lesson on Trigonometry

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### 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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