Evaluate the given integral: \[\int {x\cos xdx} \]

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Hint: The given integral can be solved by using integration by parts, that is, \[\int {udv = uv - \int {vdu} } \]. Use x as u and cos x as v to proceed with the solution.

Complete step by step answer:
Let us assign the given integral to the variable I.
\[I = \int {x\cos xdx} .......(1)\]
Integration by parts is generally used to evaluate the integrals consisting of products of functions including exponential, logarithmic, inverse trigonometric, algebraic, trigonometric functions. Here, the integral contains the product of x and cos x which is a product of algebraic and trigonometric functions.

Hence, we can solve this integral by using the formula of integration by parts.

 The formula for integration by parts is given as follows:

\[\int {udv = uv - \int {vdu} } .........(2)\]

Here, we have, u = x and dv = cos x dx.

We can find du from u, which is given as follows:

\[u = x.........(3)\]

\[du = dx.........(4)\]

We can find v by integrating dv as follows:

\[dv = \cos xdx........(5)\]

\[\int {dv} = \int {\cos xdx} \]

We know the value of integration of cos x is sin x, then we have:

\[v = \sin x.........(6)\]

Substituting equations (3), (4), (5) and (6) in equation (2), we get:

\[\int {x\cos xdx = x\sin x - \int {\sin xdx} } \]

We know that the integration of sin x is – cos x and we have the constant of integration constant C, then, we have the following:

\[\int {x\cos xdx = x\sin x - ( - \cos x)} + C\]

Simplifying the right-hand side of the expression we get:

\[\int {x\cos xdx = x\sin x + \cos x + C} \]

Hence, the value of the given integral is \[x\sin x + \cos x + C\].

Note: When using integration by parts, always use the ILATE rule to choose u. Do not forget to add the integration constant in the final answer, since it is an indefinite integral.
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