power formula

Problem
Evaluate $\displaystyle \int \left( \sqrt{x} + x\sqrt{x} + \dfrac{1}{\sqrt{x}} \right) \, dx$
 

Evaluate the following:

Example 4: $\displaystyle \int \sqrt{x^3 + 2} \,\, x^2 \, dx$

Example 5: $\displaystyle \int \dfrac{(3x^2 + 1) \, dx}{\root 3\of {(2x^3 + 2x + 1)^2}}$

Example 6: $\displaystyle \int (1 - 2x^2)^3 \, dx$
 

Evaluate the following integrals:

Example 1: $\displaystyle \int \dfrac{2x^3+5x^2-4}{x^2}dx$

Example 2: $\displaystyle \int (x^4 - 5x^2 - 6x)^4 (4x^3 - 10x - 6) \, dx$

Example 3: $\displaystyle \int (1 + y)y^{1/2} \, dy$