Integral of Sin x

Integral of sin x is -cos(x) plus a constant (C). It represents the area under the sine curve. The function repeats every 2π radians due to its periodic nature. This article explains the integral of the sine function, showing its formula, proof, and application in finding specific definite integrals. Further, it mentions solved problems and frequently asked questions.

What is Integral of Sin x?

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.

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Integral of Sin x Formula

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.

Graphical Significance of Integral of Sin x

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.

Definite Integral Method

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 ).

Geometrical Method

Consider the graph of sin(x) from ( a ) to ( b ). The area under the curve can be divided into two regions:

• Positive Area: Regions where sin(x) is positive (above the x-axis). This contributes to the positive area under the curve.
• Negative Area: Regions where sin(x) is negative (below the x-axis). This contributes to the negative area under the curve.

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:

∫₀^π/2 sin x = [-cos x]₀^π/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.

Integration of Sin x Proof by Substitution Method

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.

Definite Integral of Sin x

The definite integral of sin(x) from a to b, denoted as

∫^b_a 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.

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Integral of Sin x From 0 to π

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:

∫₀^πsin(x) dx = [-cos(π) – (-cos(0))]

∫₀^πsin(x) dx = [-(-1) + 1]

Since cos(π) is -1 and cos(0) is 1, the expression simplifies to:

∫₀^πsin(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.

Integral of Sin x From 0 to π/2

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.

Integral of Sin x – Solved Examples

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:

∫−1/2​cos(2x)dx = −1/4​sin(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:

∫1/2​sin(2x)cos(x)dx = 1/4​cos(2x)cos(x) − ∫1/4​cos(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.

Integral of Sin x – Practice Questions

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.

Integral of Sin x – Frequently Asked Questions

What is Integral of Sin x?

Integral of sin x is -cos x

What is Sin x?

Sin(x), is a trigonometry function that represents the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right-angled triangle.

What is Range of Sin x?

Range of Sin x is [-1, 1].

What is Integral and Derivative of Sin x?

The integral of sin x is -cos x and the derivative of si x is cos x

What is Integral of Sin x and Cos x?

The integral of sin x is -cos x + C and the inegral of cos x is sin x

What is Integral of Sin 2x?

The integration of sin 2x is (-cos2x)/2 + c