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
Integral of Cos x by Fundamental Theorem of Calculus
The proof of integration of cos x using Fundamental Theorem of Calculus
Let us assume
y = cos x
dy/dx = -sin x
We know that : sin2x + cos2x = 1
sin x = √(1-cos2x)
dy/dx = – √(1-cos2x)
dx = dy/-√(1-cos2x)
[ cos x = y]
dx = dy/ -√(1 – y2 )
∫cos x dx = -y.dy / √(1 – y2 )
Let us assume 1 – y2 = t
-2ydy = dt
Substituting it in integral of cos x
∫cos x dx = dt / 2√t
∫cos x dx = √t
∫cos x dx = √(1 – y2)
∫cos x dx = √(1 – cos2x)
∫cos x dx = √sin2x
∫cos x dx = sin x
Hence Proved.
Integral of Cos x by Trigonometric Substitution
The proof of cos x using trigonometric substitution is given below:
I = ∫cos x.dx
Let cos x = (eix + e-ix)/2
I = ∫(eix + e-ix)dx / 2
I = ((eix/i) + (e-ix/-i))/2 + c
I = (eix – e-ix)/2i + c
We know that
sin x = (eix – e-ix)/2i
Hence
I = sin x + c
Hence Proved.
Learn,
Integral of Cos x
Integral of Cos x is equal to Sin x + C. Integral of a function is the process of finding the area under the curve. Integration of Cos x gives the area of the region covered by 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.
Table of Content
- What is Integral of Cos x?
- Integral of Cos x Formula
- Integral of cos x Formula Proof
- Definite Integral of cos x
- Integral of Cos x – Graphical Significance
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