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**Hint:**The above integration is based on the by parts method of solving integration functions.

By parts method is the one in which is used when two functions are given to integrate simultaneously.

Using the method of integration by parts we will solve the problem before explaining the method in detail.

**Complete step by step answer:**

Let's discuss the concept of integration by parts in more details and then we will proceed for the calculation part.

When we are required to find the integration of a function which contains two functions simultaneously then, we use integration by parts concept.

We use it to examine the function and find out which of the two functions is easily integrable and which function is derivable easily. After examining we apply the formula of integration by parts which is stated below:

$\int {v\dfrac{{du}}{{dx}}} dx = uv - \int {u\dfrac{{dv}}{{dx}}} dx$

We have the function of integration which is given to us as;

$ \Rightarrow \int {{e^{\log x}}} \cos xdx$

In the above integration we have one function is ${e^{\log x}}$ and the other is $\cos x$.

We can simplify ${e^{\log x}}$as x because of the property of logarithms.

Thus, our integration will become,

$ \Rightarrow \int x \cos xdx$

Now, we will integrate $\cos x$ and differentiate x.

$ \Rightarrow \int {x\cos xdx - \int {\sin xdx} } $ (When we integrate $\cos x$we will obtain $\sin x$,then in the second integration we will again integrate $\sin x$ and differentiate x)

$ \Rightarrow x\sin x + \cos x + c$

**So, the correct answer is Option 3.**

**Note:**Integration by parts has many other applications in problem solving such as in deriving the Euler-Lagrange equation, for determining the boundary conditions in Sturm-Liouville theory, it is used in operator theory, decay of Fourier transform, Fourier transform of derivative, used in harmonics analysis, to find the gamma function identity etc.

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