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Integral of x sin x

The integral of x sin x is equal to −x cos x + sin x + C, where C is the integration constant. We can evaluate this integral using the method of integration by parts. The integral of a function gives the area under the curve of the function. sin x is one of the important trigonometric functions in trigonometry. Integration of x sin x is the process of finding the integral of x sin x, which is also called the antiderivative of x sin x as integration is the reverse process of differentiation.

Further, in this article, we will evaluate the integral of x sin using the integration by parts method of integration and derive its formula. We will also determine the definite integration of x sin x with different limits and go through some solved examples for a better understanding of the concept.

1.	What is Integral of x sin x?
2.	Integration of x sin x Formula
3.	Integral of x sin x Proof
4.	Definite Integral of x sin x
5.	FAQs on Integral of x sin x

What is Integral of x sin x?

The integral of x sin x gives the function to determine the area under the curve of the function f(x) = x sin x. The integration of x sin x is equal to −x cos x + sin x + C and it can be evaluated using the method of integration by parts (also known as the ILATE rule OR product rule of integration). Mathematically, we can write the integral of x sin x as ∫xsinx dx = −x cos x + sin x + C, where

∫ is the symbol of integration
dx shows that the integral of x sin x is with respect to the variable x
C is the constant of integration

Let us go through the formula of the integration of x sin x in the next section.

Integration of x sin x Formula

The formula for the integral of x sin x is given by, ∫xsinx dx = −x cos x + sin x + C, where C is the integration constant. We can evaluate this integral using the product rule of integration where x is the first function and sin x is the second function and x sin x is written as the product of these two functions. The image below shows the formula for the integration of x sin x.

integral of x sin x

Integral of x sin x Proof

Now that we know that the integral of x sin x is equal to −x cos x + sin x + C, we will derive this formula using the integration by parts method of integration. We use this method to find the integral of a function that is given as the product of two functions. Therefore, integration by parts is also known as the product rule of integration. Now, the formula for integration by parts is given by, ∫f(x) g(x) dx = f(x) ∫g(x) dx - ∫[f'(x) × ∫g(x) dx] dx. Here, we chose the functions f(x) and g(x) using the ILATE rule which is I - Inverse Trigonometric Function, L - Logarithmic Function, A - Algebraic Function, T - Trigonometric Function, E - Exponential Function.

Using this sequence of preference of functions, we have f(x) = x (as x is an algebraic function) and g(x) = sin x (sin x is a trigonometric function). We will also use the following formula to find the integral of x sin x:

Derivative of x: dx/dx = 1
Integral of sinx: ∫sin x dx = -cos x + C
Integral of cosx: ∫cosx dx = sinx + C

So, using the formula for the integration by parts and the above mentioned formulas, we have

∫x sinx dx = x ∫sin x dx - ∫[dx/dx × ∫sin x dx] dx

= x (-cosx) - ∫(1 × -cos x) dx

= - x cosx + ∫(1 × cos x) dx

= -x cosx + ∫cosx dx

= -x cosx + sin x + C

Hence, we have derived the formula for the integration of x sin x to be equal to -x cos x + sin x + C.

Definite Integral of x sin x

Next, in this section, we will calculate the definite integral of x sin x from 0 to π using the formula for the integral of x sin x. To determine the definite integral, we will substitute the upper limit and lower limit into the formula of integral of x sin x and subtract them. We have,

₀^π∫x sin x dx = [-x cosx + sin x + C]₀^π

= (-π cosπ + sin π + C) - (-0 cos0 + sin 0 + C)

= -π (-1) + 0 + C - 0 - 0 - C

= π

Hence, the definite integral of x sin x from 0 to π is equal to π.

Important Notes on Integral of x sin x

The integral of x sin x is equal to -x cosx + sin x + C, where C is the integration constant.
We can calculate the integral of x sin x using the method of integration by parts.
The definite integral of x sin x from 0 to π is equal to π.

☛ Related Topics:

Integral of x sin x Examples

Example 1: Evaluate the integral of x sin x^2

Solution: To find the integral of x sin(x²), we will use the substitution method of integration. Assume x² = u, then differentiating both sides, we have 2xdx = du. This implies xdx = du/2. So, we have

∫x sin(x²) dx = (1/2) ∫sin u du

= (1/2) (-cos u) + C --- [Because integral of sin x is equal to -cos x + C]

= (-1/2) cos(x²) + C

Answer: The integral of x sin x^2 is equal to (-1/2) cos(x²) + C.
Example 2: What is the integral of e^x sin x?

Solution: To find the integral of e^xsinx, we will use the integration by parts method. Using the sequence ILATE, we will assume sin x to be the first function and e^x to be the second function. So, we have

I = ∫e^xsinx dx

= sin x ∫e^xdx - ∫[d(sinx)/dx × ∫e^xdx] dx

= e^x sinx - ∫cosx e^x dx --- [Derivative of sin x is equal to cos x and integral of e^x is equal to e^x]

Applying integration by parts again to solve ∫cosx e^x dx, we have

I = e^x sinx - {cos x ∫e^x dx - ∫[d(cosx)/dx ∫e^xdx] dx}

= e^xsinx - e^xcosx + ∫-e^xsinx dx --- [Because derivative of cosx is equal to -sinx]

⇒ ∫e^xsinx dx = e^xsinx - e^xcosx - ∫e^xsinx dx

⇒ ∫e^xsinx dx + ∫e^xsinx dx = e^x (sin x - cos x)

⇒ 2 ∫e^xsinx dx = e^x (sin x - cos x)

⇒ ∫e^xsinx dx = e^x (sin x - cos x)/2 + C

Answer: ∫e^xsinx dx = e^x (sin x - cos x)/2 + C
Example 3: Calculate the integral of x sin x cos x.

Solution: First, we will simplify the expression x sin x cos x. We know that sin2A = 2sinAcosA. So, multiplying and dividing x sin x cos x by 2, we have

x sin x cos x = (2/2) x sin x cos x

= (1/2) x 2 sin x cos x

= (1/2) x sin2x

So, the integral of x sin x cos x is given by,

∫x sin x cos x dx = (1/2) ∫x sin2x dx

= (1/2) {x ∫sin2x dx - ∫[dx/dx × ∫sin2x dx] dx} --- [Applying integration by parts]

= (1/2) {(x/2) (-cos2x) - (1/2) ∫1 × -cos2x dx}

= (1/2) {(-x/2) cos2x + (1/2) ∫cos2x dx}

= (1/2) {(-x/2) cos2x + (1/4) sin2x} + C

= (-x/4) cos2x + (1/8) sin2x + C

Answer: The integral of x sin x cos x is equal to (-x/4) cos2x + (1/8) sin2x + C.

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Integral of x sin x Questions

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FAQs on Integral of x sin x

What is the Integral of x sin x in Calculus?

The integral of x sin x is equal to −x cos x + sin x + C, where C is the integration constant. Integration of x sin x is the process of finding the integral of x sin x, which is also called the antiderivative of x sin x as integration is the reverse process of differentiation

How To Find the Integral of x sinx?

We can evaluate the Integral of x sinx using the method of integration by parts. The integral of x sin x gives the function to determine the area under the curve of the function f(x) = x sin x.

What is the Integral of x sin x Formula?

The formula for the integral of x sin x is given by, ∫xsinx dx = −x cos x + sin x + C, where C is the integration constant.

What is the Definite Integration of x sin x From 0 to Pi?

The definite integral of x sin x from 0 to π is equal to π.

What is the Integral of e^x Sinx?

The integral of e^xsinx is given by, ∫e^xsinx dx = e^x (sin x - cos x)/2 + C, where C is the constant of integration.

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