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Integral of sinx is -cos(x)+C, where C is the constant of integration. This result represents the area under the sine curve and reflects the periodic nature of the sine function, which repeats every 2π radians. This article delves into the integral of the sine function, offering a detailed explanation of the formula and its derivation. It also covers the application of this integral in computing specific definite integrals. Additionally, the article includes solved problems and addresses frequently asked questions to provide a comprehensive understanding of the topic.👁 Integral-of-Sinx
Table of Content
The integral of sin(x) concerning x is -cos(x) plus a constant (C). This means that when you differentiate -cos(x) with respect to x, you get sin(x). The constant of integration (C) represents any additional constant value that may be present in the original function.
The integral of sin x physically signifies the area covered under the sine curve.
Learn,
The integral of the sine function, ∫ sin(x) dx, is equal to -cos(x) + C, where C is the constant of integration.
∫sin(x) dx = -cos(x) + C
Here, cos(x) is the cosine function, and C represents the constant that is added to the antiderivative, as the derivative of a constant is zero.
The integral of sin(x) from ( a ) to ( b ) has graphical significance in terms of calculating the area under the curve within this interval. Let's explore the graphical significance using both the definite integral method and the geometrical method.
👁 Graphical Significance of Integral of Sin x
The integral of sin(x) from ( a ) to ( b ) is given by:
= -cos(b) + cos(a)
This represents the signed area between the curve sin(x) and the x-axis from ( a ) to ( b ).
Consider the graph of sin(x) from ( a ) to ( b ). The area under the curve can be divided into two regions:
The total area is the algebraic sum of these positive and negative areas.
Example:
To find the area under the curve of sin(x) from ( a = 0 ) to ( b = π/2 ).
Using the definite integral method:
∫0π/2 sin x = [-cos x]0π/2 = -cos(π/2) - (-cos 0) = 0 + 1 = 1
This is the signed area under the curve.
Using the geometrical method:
The graph of sin(x) from 0 to (π/2) is a quarter of a circle, and the area is indeed 1.
Learn,
To find the integral of sin(x) using the substitution method, let's consider the integral:
One common substitution for trigonometric integrals involves letting u be equal to the expression inside the trigonometric function. In this case, let u = cos(x). Then, calculate du in terms of dx:
du/dx = -sin(x)
Now, solve for dx:
dx = -1/sin(x) du
Now, substitute u and dx in terms of u into the original integral:
Integral of sin(x) dx = ∫ sin(x) (-1/sin(x) du)
Simplify the expression:
Integral of sin(x) dx = -∫ du
Now integrate with respect to u:
Integral of sin(x) dx = -u + C
Now, substitute back for u, which was defined as cos(x):
Integral of sin(x) dx = -cos(x) + C
So, using the substitution method, we've arrived at the same result as in the proof by derivatives. The integral of sin(x) is -cos(x) + C, where C is the constant of integration.
Learn, Integration by Substitution
The definite integral of sin(x) from a to b, denoted as
∫ba sin(x) dx = [-cos(b) -(-cos(a)]
It calculates the net area under the sine curve between x = a and x = b, considering the direction of the area above and below the x-axis.
Learn, Definite Integral
To find the integral of sin(x) from 0 to π, we can use the antiderivative. The antiderivative of sin(x) is -cos(x). Evaluating this antiderivative from 0 to π, we get:
∫0πsin(x) dx = [-cos(π) - (-cos(0))]
∫0πsin(x) dx = [-(-1) + 1]
Since cos(π) is -1 and cos(0) is 1, the expression simplifies to:
∫0πsin(x) dx = 1 + 1 = 2
So, the integral of sin(x) from 0 to π is equal to 2. This represents the signed area between the sin(x) curve and the x-axis from x = 0 to x = π.
The definite integral represents the signed area between the curve and the x-axis over the given interval.
The integral is given as:
∫0 π/2sin(x) dx
Using the antiderivative -cos(x) to evaluate the integral:
cos(x) |[0 to π/2]
Now, substitute π/2 into -cos(x):
cos(π/2) - (-cos(0))
Recall that cos(π/2) = 0 and cos(0) = 1. Substitute these values:
-(0) - (-1)
Simplify:
0 + 1 = 1
Definite integral of sin(x) from 0 to π/2 equals 1. This means that the signed area between the sine curve and the x-axis from x = 0 to x = π/2 is 1.
Also, Check
Example 1:Find the Integral of sin2(x)
Solution:
For sin2(x), you can use the formula involving cos(2x).
∫sin2(x) dx = ∫(1 - cos(2x))/2 dx
Split it into two parts:
= (1/2)∫dx - (1/2)∫cos(2x) dx
Integral of dx is just x. The integral of cos(2x) involves using the sin(2x) formula. It looks like this:
= (1/2)x - (1/4)sin(2x) + C
Combine the two results, and add a constant "C" to account for any potential constant in the original integral.
(1/2)x - (1/4)sin(2x) + C
Example 2:Find the integral of sine3x.
Solution:
Integral of sine cubed with respect to x can be written as:
∫sin3x dx
Use a trigonometric identity to simplify:
sin3x = [1 - cos2(x)] sin(x)
∫[1 - cos2(x)] sin(x) dx
Distribute and separate the terms:
∫[sin x - sin x. cos2(x)]dx
Integrate each term separately:
-cos(x) + 1/3 cos3x + C
Here, ( C ) represents the constant of integration.
Example 3: Find integral of sin x -1
Solution:
The integral of sin(x)-1 can be expressed using the arcsine function. The integral is given by:
∫1/sin x = -ln|cosec x + cot x| + C
Here, (C) is the constant of integration.
Example 4: Find integral of sin x2
Solution:
Integral of sin²(x) with respect to x can be solved using a trigonometric identity.
∫sin2x dx = 1/2∫(1 - cos(2x)dx
Now, integrate each term separately:
1/2∫(1−cos(2x))dx = 1/2(∫1dx−∫cos(2x)dx)
= 1/2 [x - 1/2 sin(2x)] + C
where ( C ) is the constant of integration.
Example 5: Find integral of sin x -3
Solution:
Integral of sin(x)-3 with respect to (x) involves a trigonometric substitution. Here's how you can solve it:
Let u = sin(x), then du = cos(x)dx
Now, substitute these into the integral:
∫sin(x) −3dx = ∫u −3 du
Now, integrate with respect to (u):
∫u−3du = u−2/−2 + C
Substitute back in terms of (x) using u = sin(x):
∫sin(x) −3dx = -1/2sin2x + C
So, the integral of sin(x)-3 with respect to (x) is -1/2sin2x , where (C) is the constant of integration.
Example 6: Find integral of sin inverse x
Solution:
To find the integral of sin-1(x) with respect to (x), you can use integration by parts. The formula for integration by parts is:
∫udv=uv−∫vdu
u = sin-1(x) and dv = dx
Now, find (du) and (v):
v = x
Apply the integration by parts formula:
Now, integrate the remaining term on the right side. You can use substitution by letting (t = 1 - x2), then (dt = -2x , dx):
= √t + C
Now, substitute back in terms of (x):
Putting it all together:
where (C) is the constant of integration.
Example 7: Find integral of x sin 2x dx
Solution:
To find the integral of xsin(2x) with respect to (x), you can use integration by parts. The formula for integration by parts is given by:
∫udv = uv − ∫vdu
u = x and dv = sin(2x)dx
Now, find (du) and (v):
du = dx and v = -1/2cos(2x)
Apply the integration by parts formula:
∫x.sin (2x) dx = −1/2.x.cos (2x) − ∫−1/2 cos(2x) dx
Now, integrate the remaining term on the right side. The integral of -1/2cos(2x) can be found by letting (u = 2x) and using a simple substitution:
∫−1/2cos(2x)dx = −1/4sin(2x)
Substitute this result back into the original equation:
-1/2x cos(2x) + 1/4 sin(2x) + C
So, the integral of xsin(2x) with respect to (x) is -1/2x cos(2x) + 1/4 sin(2x) + C, where (C) is the constant of integration.
Example 8: Find integral of sin x cos 2x
Solution:
To find the integral of sin(x) cos(2x) with respect to (x), you can use integration by parts. The integration by parts formula is:
∫udv = uv − ∫vdu
u = sin(x) and dv = cos(2x)dx
Now, find (du) and (v):
du = cos(x) dx and v = 1/2 sin(2x)
Apply the integration by parts formula:
∫sin(x).cos(2x)dx = 1/2sin(x)sin(2x) − ∫1/2sin(2x)cos(x)dx
Now, integrate the remaining term on the right side. You can use integration by parts again:
∫1/2sin(2x)cos(x)dx = 1/4cos(2x)cos(x) − ∫1/4cos(2x)sin(x)dx
Continue the process until the integral becomes manageable. After simplifying, you will get the final result:
1/2 sin(x)sin(2x) - 1/8 cos(X) cos(2x) + 1/8 sin(X) cos(2x) + C
where (C) is the constant of integration.
Q1. Find the integral of sine from 0 to pi.
Q2. Calculate the integral of sine from -π/2 to π/2.
Q3. Find the value of the integral of sine plus cosine with respect to x.
Q4. Evaluate the integral of sine(2x) from 0 to π/3.
Q5. Find the antiderivative of sine(3x) with respect to x.
Q6. Compute the integral of sine(2x) from π to 2π.
Q7. Integrate the function sine squared with respect to x.
Q8. Evaluate the integral of sine squared from -π/4 to π/4.
The integral of sinx is -cosx+C, where C is the constant of integration. This result demonstrates the connection between the sine and cosine functions, with -cosx being the antiderivative of sinx. This integral is a fundamental concept in calculus, essential for reversing the differentiation process.In real-world applications, the integral of sinx is crucial in various fields. For example, in physics, it is used to analyze waveforms and oscillatory motion, such as sound waves and electromagnetic waves. In engineering, it helps model alternating currents and signal processing. Additionally, in fields like economics and biology, it aids in understanding periodic phenomena, such as seasonal trends and population cycles. Thus, mastering this integral not only deepens understanding of calculus but also provides valuable tools for practical problem-solving across multiple disciplines.
