Integral of (x-1)cosxdx dx

1. There are multiple ways to do this integral.

   Method #1

   1. Rewrite the integrand:

      $$\left(x - 1\right) \cos{\left(x \right)} 1 = x \cos{\left(x \right)} - \cos{\left(x \right)}$$

   2. Integrate term-by-term:

      1. Use integration by parts:

         $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

         Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$.

         Then $\operatorname{du}{\left(x \right)} = 1$.

         To find $v{\left(x \right)}$:

         1. The integral of cosine is sine:

            $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

         Now evaluate the sub-integral.

      2. The integral of sine is negative cosine:

         $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

      1. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$$

         1. The integral of cosine is sine:

            $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

         So, the result is: $- \sin{\left(x \right)}$

      The result is: $x \sin{\left(x \right)} - \sin{\left(x \right)} + \cos{\left(x \right)}$

   Method #2

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = x - 1$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$.

      Then $\operatorname{du}{\left(x \right)} = 1$.

      To find $v{\left(x \right)}$:

      1. The integral of cosine is sine:

         $$\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$$

      Now evaluate the sub-integral.

   2. The integral of sine is negative cosine:

      $$\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$$

2. Now simplify:

   $$x \sin{\left(x \right)} + \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}$$

3. Add the constant of integration:

   $$x \sin{\left(x \right)} + \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$$

The answer is:

$$x \sin{\left(x \right)} + \sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}+ \mathrm{constant}$$