integral of Xcos x

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

Alan Walker-

Published on 2023-04-21

Introduction of the integral of xcos 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 xcos x. You will also understand how to compute xcos x integral by using different integration techniques.

What is the integration of xcosx?

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

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

cos = adjacent side/hypotenuse 

Integral of xcosx formula

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

$∫x\cos xdx = x\sin x + \cos x + c$

 Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of the integral. Try integration calculator to find the solution of ∫(xcos x)dx is a single step.

How to calculate the integral xcosx?

The integral of xcos x is its antiderivative, which can be calculated using different integration techniques. In this article, we will discuss how to calculate the integral of cosine by using:

  1. Integration by parts
  2. Definite integral

Integral of xcos(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 xcosx integration by using integration by parts.

Proof of integral of xcosx by using integration by parts

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

$I=f(x)\int g(x)dx-\int[f'(x)\int g(x)dx]dx$

Suppose that, 

$f(x) = x$

$g(x) = \cos x$ 

Using these values in the by parts integration calculator

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


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

$I=x\sin x-\int \sin xdx$

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

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

Hence the integral of xcos(x) is xsin x + cos x + c, where c is a constant.

Integral of xcos x 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:

$∫^b_a f(x) dx = F(b) – F(a)$

Where a and b are known as the upper and lower bound of the integral. To calculate area between two curves for a particular integral, you use an online tool, area between two curves calculator.

Let’s understand the verification of the integral xcosx by using the indefinite integral.

Proof of integral of xcosx by using definite integral

To integrate xcosx by using a definite integral, we can use the interval from 0 to π. Let’s compute the integration of xcosx from 0 to π. For this we can write the integral as:

$∫^π_0 x\cos x dx = x\sin x+\cos x|^π_0$ 

Now, substitute the limit in the given function.

$∫^π_0 x\cos x dx =π \sin (π) +\cos (π)-(0+\cos 0)$

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

$∫^π_0 x\cos x dx = -1+(-1)=-2$

To calculate the integration of xcos x by the definite integral, you can use a tool like our definite integral calculator.


What is the integral of xcos2x?

The integral of xocs 2x can be calculated by using the method of integration by parts. It is because we have two functions multiplied with each other. Mathematically, the integral of xcos2x can be written as;

$\int x\cos 2xdx=\frac{x\sin 2x}{2}+\frac{\cos 2x}{4}+c$

What is the integration of Xsin x?

The integration of xsin x is equal to -xcos x+sin x+c, which is expressed as;

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

Where c is known as the integration constant. The integral of xsin x is calculated by using the integration by parts and the definite integral.

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