Integration of Trigonometric Functions – Formulas, Solved Examples

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
• ∫sec²x dx = tan x + C
• ∫cosec²x 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 2cos2x with respect to x.Solution- To integrate the given trigonometric functions we will use the trigonometric identity –cos2⁡x=(1+cos⁡2×2)Form this identity2cos2⁡x=1+cos⁡2xSubstituting the above value in the given integrand, we have∫2cos2⁡xdx=∫(1+cos⁡2x).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+sin⁡2×2+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:sin⁡xcos⁡y=12[sin⁡(x+y)+sin⁡(x−y)]Form this identitysin⁡4xcos⁡3x=12(sin⁡7x+sin⁡x)Therefore,∫(sin⁡4xcos⁡3x)dx=∫12(sin⁡7x+sin⁡x)dxFrom the above equation we have:∫12(sin⁡7x+sin⁡x)dx=12∫(sin⁡7x+sin⁡x)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∫(sin⁡7x)+12∫(sin⁡x)dx=−cos⁡7×14+−cos⁡x2+CThis gives us the required integration of the given function.

Example 3:

Question- Integrate sin2 x. cos2x.Solution- Before integration let us use few trigonometric relations in order to simplify the integrand.We know,  2sin⁡xcos⁡x=sin⁡2xsin⁡x.cos⁡x=sin⁡2x2Substituting the value in the given integrand, we have∫sin2⁡x.cos2⁡xdx=∫(sin⁡x.cos⁡x)2dx=∫(sin⁡2×2)2=14∫sin2⁡2x…(iAlso we know, sin2⁡x=1–cos⁡2x2Substituting the above value in equation (i), we have14∫sin2⁡2x=14∫1−cos⁡4×2=∫18dx–∫cos⁡4x8dx=18x+C1–sin⁡4×32+C2=18x–sin⁡4×32+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.

1. Find the integral of (cos x + sin x).
2. Evaluate: ∫(1 – cos x)/sin²x dx
3. Find the integral of sin²x, i.e. ∫sin²x dx.

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