Integral of x*sin2x dx

1. There are multiple ways to do this integral.

   Method #1

   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 $\operatorname{dv}{\left(x \right)} = \sin{\left(2 x \right)}$.

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

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

      1. There are multiple ways to do this integral.

         Method #1

         1. Let $u = 2 x$.

            Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

            $$\int \frac{\sin{\left(u \right)}}{2}\, du$$

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

               $$\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$$

               1. The integral of sine is negative cosine:

                  $$\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$$

               So, the result is: $- \frac{\cos{\left(u \right)}}{2}$

            Now substitute $u$ back in:

            $$- \frac{\cos{\left(2 x \right)}}{2}$$

         Method #2

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

            $$\int 2 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$$

            1. Let $u = \cos{\left(x \right)}$.

               Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$:

               $$\int \left(- u\right)\, du$$

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

                  $$\int u\, du = - \int u\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u\, du = \frac{u^{2}}{2}$$

                  So, the result is: $- \frac{u^{2}}{2}$

               Now substitute $u$ back in:

               $$- \frac{\cos^{2}{\left(x \right)}}{2}$$

            So, the result is: $- \cos^{2}{\left(x \right)}$

      Now evaluate the sub-integral.

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

      $$\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$$

      1. Let $u = 2 x$.

         Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

         $$\int \frac{\cos{\left(u \right)}}{2}\, du$$

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

            $$\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$$

            1. The integral of cosine is sine:

               $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

            So, the result is: $\frac{\sin{\left(u \right)}}{2}$

         Now substitute $u$ back in:

         $$\frac{\sin{\left(2 x \right)}}{2}$$

      So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$

   Method #2

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

      $$\int 2 x \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int x \sin{\left(x \right)} \cos{\left(x \right)}\, dx$$

      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 $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)} \cos{\left(x \right)}$.

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

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

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

            $$\int \frac{\sin{\left(2 x \right)}}{2}\, dx = \frac{\int \sin{\left(2 x \right)}\, dx}{2}$$

            1. Let $u = 2 x$.

               Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

               $$\int \frac{\sin{\left(u \right)}}{2}\, du$$

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

                  $$\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$$

                  1. The integral of sine is negative cosine:

                     $$\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$$

                  So, the result is: $- \frac{\cos{\left(u \right)}}{2}$

               Now substitute $u$ back in:

               $$- \frac{\cos{\left(2 x \right)}}{2}$$

            So, the result is: $- \frac{\cos{\left(2 x \right)}}{4}$

         Now evaluate the sub-integral.

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

         $$\int \left(- \frac{\cos{\left(2 x \right)}}{4}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{4}$$

         1. Let $u = 2 x$.

            Then let $du = 2 dx$ and substitute $\frac{du}{2}$:

            $$\int \frac{\cos{\left(u \right)}}{2}\, du$$

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

               $$\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$$

               1. The integral of cosine is sine:

                  $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

               So, the result is: $\frac{\sin{\left(u \right)}}{2}$

            Now substitute $u$ back in:

            $$\frac{\sin{\left(2 x \right)}}{2}$$

         So, the result is: $- \frac{\sin{\left(2 x \right)}}{8}$

      So, the result is: $- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{4}$

2. Add the constant of integration:

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

The answer is:

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