# Integral Of Cos(Lnx)

Integral of cos(lnx) along with its formula and proof with examples. Also learn how to calculate integration of cos(lnx) with step by step examples.

Alan Walker-

Published on 2023-04-14

## Introduction to integral of cos(lnx)

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 cos(ln x) integral by using different integration techniques.

## What is the integral of cos(ln x)?

The integral of cos(lnx) is an antiderivative of cos(ln x) function which is equal to ½[xsin(ln x) + xcos(ln x)]. It is also known as the reverse derivative of sine function which is a trigonometric identity.

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

cos = adjacent side / hypotenuse

### Integral of cos(ln x) formula

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

$\int \cos(\ln x)dx = \frac{1}{2}[x\sin(\ln x) + x\cos(\ln x)]+c$

Where c is any constant involved, dx is the coefficient of integration and ∫ is the symbol of integral. If we replace cos(ln x) by cos(e^x) in the above integration formula, we can calculate the integral of cos(e^x)

## How to calculate the cos(ln x) integral?

The integral of cos(lnx) is its antiderivative that can be calculated by using different integration techniques. In this article, we will discuss how to calculate integral of cosine by using:

1. Integration by parts
2. Substitution method
3. Definite integral

## Integral cos(lnx) 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 integral of two functions combined together. Let’s discuss calculating the integral of cos ln x by using integration by parts.

### Proof of integral of cos(lnx)dx by using integration by parts

Since we know that the function sine squared x can be written as the product of two functions. Therefore, we can integrate cos(lnx) by using the integration by parts. For this, suppose that:

$I = \cos(\ln x)$

Applying the integral we get,

$I =\int \cos(\ln x))dx$

Since the formula of integration by parts calculator is:

$\int[f(x).g(x)]dx = f(x).\int g(x)dx - \int[f’(x).\int g(x)]dx$

Now replacing f(x) and g(x) by sin(lnx)(1/x) and x, we get,

$I = x.\cos(\ln x) + \int \frac{x\sin(\ln x)}{x}dx$

It can be written as:

$I = x.\cos(\ln x) – \int[\sin(\ln x)]dx$

Now by using integration by parts again,

$I = x.\cos(\ln x) + [x\sin(\ln x) - \int \cos(\ln x)dx]$

Since

$I = ∫\cos(\ln x)dx$

Therefore,

$I = x.\cos(\ln x) + [x\sin(\ln x) - I]$

Moreover,

$2I = x.\cos(\ln x) + x\sin(\ln x)$

Hence the integral of sin(ln x) by using integration by parts is: