Integral of 6^(sin(x))*cos(x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = 6^{\sin{\left(x \right)}}$.

      Then let $du = 6^{\sin{\left(x \right)}} \log{\left(6 \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{\log{\left(6 \right)}}$:

      $$\int \frac{1}{\log{\left(6 \right)}^{2}}\, du$$

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

         $$\int \frac{1}{\log{\left(6 \right)}}\, du = \frac{\int 1\, du}{\log{\left(6 \right)}}$$

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

            $$\int 1\, du = u$$

         So, the result is: $\frac{u}{\log{\left(6 \right)}}$

      Now substitute $u$ back in:

      $$\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}$$

   Method #2

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

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

      $$\int 6^{u}\, du$$

      1. The integral of an exponential function is itself divided by the natural logarithm of the base.

         $$\int 6^{u}\, du = \frac{6^{u}}{\log{\left(6 \right)}}$$

      Now substitute $u$ back in:

      $$\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}$$

2. Add the constant of integration:

   $$\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}+ \mathrm{constant}$$

The answer is:

$$\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}+ \mathrm{constant}$$