Sample Problems on Integration of Trigonometric Functions

Problem 1: Determine the integral of the following function: f(x) = cos³ x.

Solution:

Let us consider the integral of the given function as,

I = ∫ cos³ x dx

It can be rewritten as:

I = ∫ (cos x) (cos²x) dx

Using trigonometry identity; cos²x = 1 – sin²x, we get

I = ∫ (cos x) (1 – sin²x) dx

⇒ I = ∫ cos x – cos x sin²x dx 

⇒ I = ∫ cosx dx – ∫ cosx sin²x dx

As ∫ cos x dx = sin x + C,

Thus, I = sin x – ∫ sin²x cos x dx                         . . . (1)

Let, sin x = t 

⇒ cos x dx = dt.

Substitute t for sin x and dt for cos x dx in second term of the above integral.

I = sin x – ∫ t² dt

⇒ I = sin x – t³/3 + C

Again, substitute back sin x for t in the expression.

Hence, ∫ cos³x dx = sin x – sin³ x / 3 + C.

Problem 2: If f(x) = sin² (x) cos³ (x) then determine ∫ sin²(x) cos³(x) dx.

Solution:

Let us consider the integral of the given function as,

I = ∫ sin²(x) cos³(x) dx

Using trigonometry identity; cos² x = 1 – sin² x, we get

I = ∫ sin² x (1 – sin² x) cos x dx

Let sin x = t then,

⇒ dt = cos x dx

Substitute these in the above integral as,

I = ∫ t² (1 – t²) dt

⇒ I = ∫ t² – t⁴ dt

⇒ I = t³ / 3 – t⁵ / 5 + C  

Substitute back the value of t in the above integral as,

Hence, I = sin³ x / 3 – sin⁵ x / 5 + C.

Problem 3: Let f(x) = sin⁴(x) then find ∫ f(x)dx. i.e. ∫ sin⁴(x) dx.

Solution:

Let us consider the integral of the given function as,

I = ∫ sin⁴(x) dx

⇒ I = ∫ (sin²(x))² dx

Use\ing trigonometry identity; sin²(x) = (1 – cos (2x)) / 2, we get

I = ∫ {(1 – cos (2x)) / 2}² dx

⇒ I = (1/4) × ∫ (1+cos²(2x)- 2 cos2x) dx

⇒ I = (1/4) × ∫ 1 dx + ∫ cos²(2x) dx – 2 ∫ cos2x dx

⇒ I = (1/4) × [ x + ∫ (1 + cos 4x) / 2 dx – 2 ∫ cos2x dx ] 

⇒ I = (1/4) × [ 3x / 2 + sin 4x / 8 – sin 2x ] + C

⇒ I = 3x / 8 + sin 4x / 32 – sin 2x / 4 + C

Hence, ∫ sin⁴(x) dx = 3x / 8 + sin 4x / 32 – sin 2x / 4 + C

Problem 4: Find the integration of [Tex]\bold{\int\frac{e^{tan^{-1}x}}{1+x^2} dx} [/Tex].

Solution:

Let us consider the integral of the given function as,

[Tex]I =\int \frac{e^{tan^{-1}x}}{1+x^2} dx [/Tex]

Let t = tan^-1 x                         . . . (1)

Now, differentiate both side with respect to x:

dt = 1 / (1+x²) dx

Therefore, the given integral becomes:

I = ∫ e^t dt

⇒ I = e^t + C                               . . . (2)

Substitute the value of (1) in (2) as:

⇒  [Tex]I = e^{tan^{-1}x} + C [/Tex]

Which is the required integration for the given function.

Problem 5: Find the integral of the function f (x) defined as,

f(x) = 2x cos (x² – 5) dx

Solution:

Let us consider the integral of the given function as,

I = ∫ 2x cos (x² – 5) dx

Let (x² – 5) = t                         . . . (1)

Now differentiate both side with respect to x as,

2x dx = dt 

Substituting these values in the above integral,

I = ∫ cos (t) dt

⇒ I = sin t + C                         . . . (2)

Substitute the value equation (1) in equation (2) as,

⇒ I = sin (x² – 5) + C

This is the required integration for the given function. 

Problem 6: Determine the value of the given indefinite integral, I = ∫ cot (3x +5) dx.

Solution:

The given integral can be written as,

I = ∫ cot (3x +5) dx 

⇒ I = ∫ cos (3x +5) / sin (3x +5) dx 

Let, t = sin(3x + 5)

⇒ dt = 3 cos (3x+5) dx

⇒ cos (3x+5) dx = dt / 3 

Thus,

I = ∫ dt / 3 sin t  

⇒ I = (1 / 3) ln | t | + C

Replace t by sin (3x+5) in the above expression.

I = (1 / 3) ln | sin (3x+5) | + C

This is the required integration for the given function.

Integration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.

Below is an example of the Integration of a given function.

e.g., Consider a function, f(y) = y². 

This function can be integrated as: 

∫y²dy = [Tex]\frac{y^{2+1}}{2+1}~+~C[/Tex]

However, an indefinite integral is a function that takes the anti-derivative of another function. It is represented as an integral symbol (∫), a function, and a derivative of the function at the end. The indefinite integral is an easier way to symbolize an anti-derivative.

Let’s learn what is integration mathematically, 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 of 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 and x is the variable of integration.