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Integral of Cos x
Before going to find the integral of cos x, let us recall what is integral. An integral is nothing but the antiderivative. Antiderivative, as its name suggests, can be found by using the reverse process of differentiation. i.e.,
 Differentiation is finding f'(x) from f(x).
 Integration is finding f(x) from f'(x).
Thus, the integration of cos x is found by using differentiation. Let us see more about the integral of cos x along with its formula and proof (in different methods).
What is the Integral of cos x?
The integral of cos x dx is sin x. Mathematically, this is written as ∫ cos x dx = sin x + C, where, C is the integration constant. Here,
 '∫' is a symbol of integration and it is known as the "integral"
 cos x is the integrand
 dx means the integration is with respect to x.
But how to do the integration of cos x? We are going to find the integral of cos x in various methods such as using the derivatives and using the substitution method in the upcoming sections.
Proof of Integral of Cos x by Derivatives
The integration is the opposite (reverse process) of differentiation. So to find the integral of cos x, we have to see by differentiating what function would give us cos x. i.e., we have to see
d/dx \(\fbox{?}\) = cos x.
Let us recall the formulas of differentiation and search for some formula that gives us cos x as the derivative. Yes, we found one formula which says d/dx (sin x) = cos x.
Thus, the derivative of sin x is cos x. So the integration of cos x (antiderivative) must be sin x. We have to add an integration constant after integrating any function. Thus, we have
∫ cos x dx = sin x + C
Hence proved.
Proof of Integral of Cos x by Substitution Method
We can prove that the integral of cos x to be sin x + C using the substitution method. For this, we assume that y = cos x. Then dy/dx = sin x (or)
dy = sin x dx
By trigonometric identities, sin x = √1  cos²x. Then the above equation becomes,
dy = √1  cos²x dx
dy = √1  y² dx
dy / √1  y² = dx
Multiplying both sides by cos x,
(cos x dy) / √1  y² = cos x dx
Again substitute cos x = y on the left side.
(y dy) / √1  y² = cos x dx
Integrating on both sides,
∫ (y dy) / √1  y² = ∫ cos x dx
Let 1  y² = u. Then 2y dy = du (or) y dy = 1/2 du.
Then the above lefthand side integral becomes,
(1/2) ∫ 1/√u du = ∫ cos x dx
(1/2) ∫ u^{1/2 }du = ∫ cos x dx
Using one of the integration formulas, ∫ x^{n} dx = (x^{n+1})/(n+1) + C. So we get
(1/2) (u^{1/2} / (1/2)) + C = ∫ cos x dx
u^{1/2} + C = ∫ cos x dx
Substituting u = 1  y² back here,
(1  y²)^{1/2} + C = ∫ cos x dx
Substitute y = cos x back here,
(1  cos²x)^{1/2} + C = ∫ cos x dx
(sin²x)^{1/2} + C = ∫ cos x dx
sin x + C = ∫ cos x dx
Hence proved.
Definite Integral of Cos x
We know that ∫ cos x dx = sin x + C. So by the fundamental theorem of calculus, ∫ₐ^{ᵇ} cos x dx = [sin b + C]  [sin a + C] = sin b + C  sin a  C = sin b  sin a. Notice that the integration constant C is NOT present in the definite integral of cos x. So we always ignore C while calculating the definite integral. Let us see some examples.
 ∫₀^{π} cos x dx = [sin x]₀^{π}
= sin π  (sin 0)
= 0  0
= 0  ∫₀^{π/2} cos x dx = [sin x]₀^{π/2}
= sin π/2  (sin 0)
= 1  0
= 1
Graphical Intuition of Integral of Cos x
The integral of a function within an interval is nothing but the area occupied by its graph in that interval. Let us now graph the function f(x) = cos x and calculate the approximate area under the curve for some intervals by using geometric formulas. Also, we will calculate the exact areas by using the definite integral of cos x within the same intervals and compare both results.
Now, we will calculate the areas under the curve of y = cos x in each of the intervals [0, π/2] and [0, π]. To calculate the approximate areas, we drew the triangles. They don't give the exact area. From the figure, it is clear that the area calculated like using the triangles is less than the actual area as the triangle(s) doesn't cover the complete area under the curve.
Interval  Manual Area  Area by Definite Integral 
[0, π/2]  Area of triangle = 1/2 × π/2 × 1 = π/4 ≈ 0.8 
∫₀^{π/2} cos x dx 
[0, π]  Area of triangle = (1/2 × π/2 × 1)  (1/2 × π/2 × 1) = π/4  π/4 ≈ 0 
∫₀^{π} cos x dx = [sin x]₀^{π} = 0 
The numbers in the second column and third column are (approximately) equal (anyway, the numbers in the second column are less than the numbers of the third column as the triangles are NOT covering the entire area). Thus, the integral of cos x is sin x and is geometrically proved.
Topics Related to Integral of Cos x:
Here are some topics that you may be interested in while doing integration of cos x.
Solved Examples Using Integration of Cos x

Example 1: Find the value of ∫ cos 2x dx.
Solution:
Let 2x = u. Then 2 dx = du (or) dx = du/2
Then the given integral becomes,
= ∫ cos u (du/2)
= (1/2) ∫ cos u du
We know that the integral of cos x is sin x + C. So the value of the above integral is,
= (1/2) [sin u] + C
= (sin 2x)/2 + C (as u = 2x)
Answer: ∫ cos 2x dx = (sin 2x)/2 + C

Example 2: Find the value of ∫ x cos (x^{2}) dx.
Solution:
The given integral can be written as ∫ cos (x^{2}) (x dx).
Let x^{2} = u. Then 2x dx = du (or) x dx = (1/2) du.
Substituting these values, the above integral becomes,
∫ cos u (1/2) du = (1/2) (sin u) + C (as the integration of cos x is sin x)
= (1/2) (sin x^{2}) + C
Answer: ∫ x cos (x^{2}) dx = (sin x^{2})/2+C

Example 3: Evaluate the integral ∫ (cos x) / x dx.
Solution:
The series expansion of cos x = 1  x^{2}/2! + x^{4}/4!  x^{6}/6! + ...
(cos x)/ x = [1  x^{2}/2! + x^{4}/4!  x^{6}/6! + ...] / x
= 1/x  x/2! + x^{3}/4!  x^{5}/6! + ...
Now we will take the integral on both sides,
∫ (cos x) / x dx
= ∫ [1/x  x/2! + x^{3}/4!  x^{5}/6! + ...] dx
We have ∫ x^{n} dx = (x^{n+1})/(n+1) + C. Also, ∫ 1/x dx = ln x + C. Using these,
∫ (cos x) / x dx = ln x  x^{2}/(2 · 2!) + x^{4}/(4 · 4!)  x^{6}/(6 · 6!) + ... + C
Answer: The value of the integral cos x/x is ln x  x^{2}/(2 · 2!) + x^{4}/(4 · 4!)  x^{6}/(6 · 6!) + ... + C
FAQs on Integral of Cos x
What is the Integral of Cos x?
The integral of cos x is sin x + C. i.e., ∫ cos x dx = sin x + C. Here, C is the integration constant.
Is Integral of Cos x Equal to  Sin x?
No, the integral of cos x is NOT sin x, in fact, the integration of cos x is sin x + C.
How to Find Integral of Cos x?
We have d/dx (sin x) = cos x. Therefore, the integral of cos x, being the antiderivative of cos x, is sin x + C.
What is the Integral of Cos x From 0 to 2π?
We know that ∫ cos x dx = sin x. If we apply the limits 0 and 2π, we get sin 2π  (sin 0) = 0  0 = 0.
Why is the Integral of Cos x Equal to Sin x?
We are aware that the derivative of sin x is cos x. Since integral is nothing but antiderivative, the integral of cos x is sin x (of course, we add the integration constant C to this).
What is the Antiderivative of Cos x?
The antiderivative of cos x is nothing but the integration of cos x and hence it is equal to sin x + C.
What is the Integral of Cos 3x?
Let 3x = u then 3 dx = du. From this, dx = (1/3) du. So ∫ cos 3x dx = ∫ cos u (1/3) du = (1/3) (sin u) + C. Substituting u = 3x back, we get ∫ cos 3x dx = (1/3) (sin (3x)) + C.
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