Integral of Xsin x

Integral of xsin x along with its formula and proof with examples. Also learn how to calculate integration of xsin x with step by step examples.

Alan Walker-

Published on 2023-04-21

Introduction to the integral of xsin x

In calculus, the integral is a fundamental concept that assigns numbers to functions to define displacement, area, volume, and all those functions that contain a combination of tiny elements. It is categorized into two parts, definite integral and indefinite integral. The process of integration calculates the integrals. This process is defined as finding an antiderivative of a function. 

Integrals can handle almost all functions, such as trigonometric, algebraic, exponential, logarithmic, etc. This article will teach you what is integral to a trigonometric function sine. You will also understand how to compute xsinx integral by using different integration techniques.

What is the integral of xsinx?

The integral of xsin x is an antiderivative of sine function which is equal to –xcos x + sin x+c. It is also known as the reverse derivative of sine function which is a trigonometric identity. 

The sine function is the ratio of opposite side to the hypotenuse of a triangle which is written as:

Sin = opposite side / hypotenuse 

Integral of xsin x formula

The formula of integral of sin contains integral sign, coefficient of integration and the function as sine. It is denoted by ∫(xsin x)dx. In mathematical form, the integral of x sinx is:

$∫x\sin xdx = -x\cos x + \sin x+ c$

 Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. 

How to calculate the integration of xsinx?

The integral of xsin x is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of sine by using:

  1. Integration by parts
  2. Definite integral

Integral of xsin(x) by using integration by parts

The derivative of a function calculates the rate of change, and integration is the process of finding the antiderivative of a function. The integration by parts is a method of solving the integral of two functions combined together. Let’s discuss calculating the integral of x sin x by using integration by parts.

Proof of integral of xsin x by using integration by parts

To integrate xsinx using integration by parts, we can use the following formula:

$I=f(x)\int g(x)dx-\int[\frac{d}{dx}(f(x))\int g(x)dx]dx$

Suppose that, 

$f(x) = x$

$g(x) = \sin x$ 

Using these values in the integration by parts calculator

$I=x\int \sin xdx-\int [\frac{d}{dx}(x)\int \sin xdx]dx$


$I=-x\cos x-\int(1)(-\cos x)dx$

$I=-x\cos x+\int \cos xdx$

Since the integral of cos x is equal to sin x, then 

$I=-x\cos x+\sin x+c$

Hence the integration of xsinx is -xcos x + sin x + c, where c is a constant.

Integral of xsinx by using definite integral 

The definite integral is a type of integral that calculates the area of a curve by using infinitesimal area elements between two points. The definite integral can be written as:

$∫^b_a f(x) dx = F(b) – F(a)$

Let’s understand the verification of the integral of x sin x by using the indefinite integral calculator.

Proof of integral of xsin(x) by using definite integral

To integrate xsinx by using a definite integral, we can use the interval from 0 to π or 0 to π/2. Let’s compute the integral of xsin x from 0 to π. For this we can write the integral as:

$\int^{\pi}_0 x\sin xdx=-x\cos x+\sin x|^{\pi}_0$

Now, substitute the limit in the given function.

$\int^{\pi}_0 x\sin xdx=(-\pi \cos \pi+\sin \pi)-(-0+0)$

Since sin pi is equal to 0 and cos π is equal to -1, therefore, 

$\int^{\pi}_0 x\sin xdx=-\pi (-1)+0=\pi$

Which is the calculation of the definite integral of xsin x. You can also try our integral calculator with steps to make your calculations quick and easy.

Frequently Asked Questions

What is the integration of xsinx from 0 to infinity?

The integration or integral of xsinx from 0 to infinity is equal to pi. It can be calculated by using the definite integral formula which is used to evaluate integrals bounded on a specific interval.

How do you integrate Xsin 2x?

You can easily calculate the integral of xsin 2x by using integration by parts rule because it is the same as the integral of xsin x. The calculations of the integral of xsin 2x is,

$\int x\sin 2xdx=-2x\cos 2x+\sin 2x+c$

Where the integral of sin 2x is equal to -2cos 2x.

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