Integral of Cos x

Integral of Cos x is Sin x. Integral is the process of finding the area under the curve. Integration of Cos x gives the idea of the region covered under the cosine trigonometric function. The integral also called the Antiderivative of a function exists when the function is differentiable. Integration of Cos x is possible as the cosine function is also differentiable in its domain.

In this article, we will learn what is Integral of cos x, the formula of the Integral of cos x, and how to integrate cos x.

What is Integral of Cos x?

Integral of cos x. Integration of Cosine exists as cosine is a differentiable function. The derivative of Differentiation and Integration are reverse processes. When we know the derivate of a function, we can derive the function with the help of integration but with a constant as the constants are eliminated while finding out the derivate because derivative of a constant is 0. So we add a constant ‘c’ whenever we integrate. One of the applications of Integration is it is used to find area under the curve.

Integral of cos x gives information about the area under the curve of cos x and its definite integral gives area under the graph of cos x within specified bounds.

Learn, Cosine Function

Integral of Cos x Formula

The integral of Cos x is sin x. Hence, the formula of integral of cos x is given as:

Since, d(sin x)/dx = cos x

∫cos x dx = sinx + c

Learn, Integration Formulas

Integral of Cos x Formula Proof

The Integral of Cos x Formula can be derived using following two ways:

• By Fundamental Theorem of Calculus
• By Trigonometric Substitution

1. By Fundamental Theorem of Calculus

The proof of integration of cos x using Fundamental Theorem of Calculus

Let us assume

y = cosx

dy/dx = -sinx

We know that : sin²x + cos²x = 1

sinx = √(1-cos²x)

dy/dx = – √(1-cos²x)

dx = dy/-√(1-cos²x)

[ cosx = y]

dx = dy/ -√(1 – y² )

∫cosx dx = -y.dy / √(1 – y² )

Let us assume 1 – y² = t

-2ydy = dt

Substituting it in integral of cosx

∫cosx dx = dt / 2√t

∫cosx dx = √t

∫cosx dx = √(1 – y²)

∫cosx dx = √(1 – cos²x)

∫cosx dx = √sin²x

∫cosx dx = sinx

Hence Proved.

2. By Trigonometric Substitution

The proof of cos x using trigonometric substitution is given below:

I = ∫cos x.dx

Let cos x = (e^ix + e^-ix)/2

I = ∫(e^ix + e^-ix)dx / 2

I = ((e^ix/i) + (e^-ix/-i))/2 + c

I = (e^ix – e^-ix)/2i + c

We know that

sinx = (e^ix – e^-ix)/2i

Hence

I = sin x + c

Hence Proved.

Learn,

• Fundamental Theorem of Calculus
• Integration by Substitution

Definite Integral of cos x

Definite Integral of cosx gives area under the graph of cosx with in given bounds [a,b]. It is given by ∫_a^b cosxdx where a and b are the limits of integration.

Definite Integral formula

∫_a^b f(x) dx = F(b) – F(a)

Definite Integral of cos x

∫_a^b cosxdx = [sinx]_a^b= sinb – sina

Definite integral of cos x from 0 to π/2

∫^π/2₀cosxdx = [sinx] ^π/2₀

= sin π/2 – sin 0

= 1 – 0

= 1

Definite integral of cosx from π/2 to π

∫^π_π/2cosxdx = [sinx]^π_π/2

= sinπ – sinπ /2

= 0 – 1

= -1

Integral of Cos x – Graphical Significance

We know that the integration gives the idea about the area under the curve. Hence, the integral of cos x also gives the area under the cosine curve within a defined range. The area under the cosine curve is shown below:

Area under Cosine Curve from 0 to π/2

Approximate calculation of Area under cosine curve is given as

Area of Triangle = 1/2 × b × h = 1/2 × π/2 × 1 = π/4 ≈ 0.8

Area under cosine curve from 0 to π/2 using integration is given as:

Area = ∫^π/2₀cosxdx = [sinx] ^π/2₀

= sin π/2 – sin 0

= 1 – 0

= 1

Area under Cosine Curve from 0 to π

Approximate calculation of Area under Cosine Curve is given as:

Area of Triangle 1 + Area of Triangle 2 = (1/2 × π/2 × 1) – (1/2 × π/2 × 1) = 0

Area under Cosine curve using Integration is given as:

Area = ∫^π₀cosxdx = [sinx] ^π₀

= sin π – sin 0

= 0 – 0

= 0

Hence, we verified that the integration of cos x gives the area under the cosine curve under the defined limits

Integral of Cos x – Solved Examples

Example 1. I = ∫2cosx dx/3sin²x . Evaluate I.

Solution:

I = 2/3 ∫cosec x.cot x dx

I = (-2cosec x/3) + c

Example 2. Find the integral of sinx.cosx

Solution:

I = ∫(2sinx.cosx)/2

I = ∫sin2x/2

I = -cos2x/4 + c

Example 3. Find ∫(sin²x – cos²x).dx/(sin x cos x)

Solution:

I = ∫(sin²xdx/sin x.cos x) – ∫(cos²xdx/sin x.cos x)

I = ∫tan x dx – ∫cot x dx

= log|secx|-[-log|cosecx|] + c

= log|secx| + log|cosecx| + c

Example 4. Find ∫(sin²x – cos²x)dx / sin²x*cos²x.

Solution:

I = ∫sec²xdx – ∫cosec²xdx

= tanx + cotx + c

Example 5. Find ∫sin⁶xdx/cos⁸x.

Solution:

I = ∫sin⁶xdx/cos⁸x

= ∫tan⁶x.sec²xdx

Let tanx = t

sec²x dx = dt

I = ∫t⁶ dt

I = t⁷/7 + c

= tan⁷x/7 + c

Example 6. Find ∫(cosx)²dx.

Solution:

cos2x = 2cos²x – 1

I = ∫(cos2x+1)dx/2

= (sin2x)/(2*2) + (x/2) + c

= sin2x/4 + x/2 + c

Example 7. Find ∫(cosx)^-1 dx.

Solution:

By using ILATE

I = ∫1.cos^-1x.dx

= cos^-1x.x + ∫(-1)/(√1-x²) * xdx

= xcos^-1x + (1/2) * ∫(2x dx)/(√(1-x²)

Let 1 – x² = t

-2xdx = dt

= xcos^-1x -(1/2)*t^1/2/(1/2) + c

= xcos^-1x – √t + c

= xcos^-1x – √(1 – x²) + c

Example 8. Evaluate I = ∫(cos2x – cos2a)dx/(cosx – cosa).

Solution:

I = ∫(cos2x – cos2a)dx/(cosx-cosa)

We know that cos2x = 2cos²x – 1

= ∫((2cos²x – 1) – (2cos²a – 1))dx/(cosx – cosa)

= ∫(2(cos²x – cos²a))dx/(cosx – cosa)

= ∫(2(cosx + cosa)(cosx – cosa)dx/(cosx – cosa)

= ∫2(cosx + cosa)dx

= 2(sinx + xcosa) + c

Integral of Cos x – Practice Problems

Try out the following practice problems on Integration of Cos x

Q1. Evaluate ∫dx/sin²x * cos²x.

Q2. Find the integral of cos^-1(sinx).

Q3. Find ∫2.dx/(1 + cos2x).

Q4. Find the value of ∫cos3x.

Q5. Evaluate ∫dx/(sin⁴x + cos⁴x).

Integral of Cos x – FAQs

What is Integral of Cos x?

The integration of cos x is sin x

What is Integral of Sec x?

The integration of sec x is ∫secx = log|secx + tanx| + c

What is Integral of Sec²x?

The integral of sec²x is ∫sec²xdx = tanx + c

What is Integral of sin x cos x?

The integral of sin x cos x is (-1/4) cos 2x + C. This can be obtained as follows

Multiplying and dividing sin x cos x by 2

I = (∫2 sinx cosx dx)/2

= ∫sin 2x dx /2

= (-1/4) cos 2x+ c

What is Integral of cos 2x?

The integral of cos 2x is (1/2)Sin 2x + C

What is Integral of cos 4x?

The integral of cos x is (1/4)sin 4x + C