Integral of cosx(2sinx+4) dx

1. There are multiple ways to do this integral.

   Method #1

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

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

      $$\int \left(2 u + 4\right)\, du$$

      1. Integrate term-by-term:

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

            $$\int 2 u\, du = 2 \int u\, du$$

            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: $u^{2}$

         1. The integral of a constant is the constant times the variable of integration:

            $$\int 4\, du = 4 u$$

         The result is: $u^{2} + 4 u$

      Now substitute $u$ back in:

      $$\sin^{2}{\left(x \right)} + 4 \sin{\left(x \right)}$$

   Method #2

   1. Rewrite the integrand:

      $$\left(2 \sin{\left(x \right)} + 4\right) \cos{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \cos{\left(x \right)}$$

   2. Integrate term-by-term:

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

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

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

         1. The integral of cosine is sine:

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

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

      The result is: $4 \sin{\left(x \right)} - \cos^{2}{\left(x \right)}$

   Method #3

   1. Rewrite the integrand:

      $$\left(2 \sin{\left(x \right)} + 4\right) \cos{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \cos{\left(x \right)}$$

   2. Integrate term-by-term:

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

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

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

         1. The integral of cosine is sine:

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

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

      The result is: $4 \sin{\left(x \right)} - \cos^{2}{\left(x \right)}$

2. Now simplify:

   $$\left(\sin{\left(x \right)} + 4\right) \sin{\left(x \right)}$$

3. Add the constant of integration:

   $$\left(\sin{\left(x \right)} + 4\right) \sin{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\left(\sin{\left(x \right)} + 4\right) \sin{\left(x \right)}+ \mathrm{constant}$$