Integral of sec(5x)tan(5x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \sec{\left(5 x \right)}$.

      Then let $du = 5 \tan{\left(5 x \right)} \sec{\left(5 x \right)} dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{1}{25}\, du$$

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

         $$\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}$$

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

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

         So, the result is: $\frac{u}{5}$

      Now substitute $u$ back in:

      $$\frac{\sec{\left(5 x \right)}}{5}$$

   Method #2

   1. Let $u = 5 x$.

      Then let $du = 5 dx$ and substitute $\frac{du}{5}$:

      $$\int \frac{\tan{\left(u \right)} \sec{\left(u \right)}}{25}\, du$$

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

         $$\int \frac{\tan{\left(u \right)} \sec{\left(u \right)}}{5}\, du = \frac{\int \tan{\left(u \right)} \sec{\left(u \right)}\, du}{5}$$

         1. The integral of secant times tangent is secant:

            $$\int \tan{\left(u \right)} \sec{\left(u \right)}\, du = \sec{\left(u \right)}$$

         So, the result is: $\frac{\sec{\left(u \right)}}{5}$

      Now substitute $u$ back in:

      $$\frac{\sec{\left(5 x \right)}}{5}$$

2. Add the constant of integration:

   $$\frac{\sec{\left(5 x \right)}}{5}+ \mathrm{constant}$$

The answer is:

$$\frac{\sec{\left(5 x \right)}}{5}+ \mathrm{constant}$$