Integral of (3/sqrt(x))+4/(x^8) dx

1. Integrate term-by-term:

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

      $$\int \frac{4}{x^{8}}\, dx = 4 \int \frac{1}{x^{8}}\, dx$$

      1. Don't know the steps in finding this integral.

         But the integral is

         $$- \frac{1}{7 x^{7}}$$

      So, the result is: $- \frac{4}{7 x^{7}}$

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

      $$\int \frac{3}{\sqrt{x}}\, dx = 3 \int \frac{1}{\sqrt{x}}\, dx$$

      1. Let $u = \sqrt{x}$.

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

         $$\int 2\, du$$

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

            $$\text{False}$$

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

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

            So, the result is: $2 u$

         Now substitute $u$ back in:

         $$2 \sqrt{x}$$

      So, the result is: $6 \sqrt{x}$

   The result is: $6 \sqrt{x} - \frac{4}{7 x^{7}}$

2. Now simplify:

   $$\frac{2 \left(21 x^{\frac{15}{2}} - 2\right)}{7 x^{7}}$$

3. Add the constant of integration:

   $$\frac{2 \left(21 x^{\frac{15}{2}} - 2\right)}{7 x^{7}}+ \mathrm{constant}$$

The answer is: