Question

Alguien que me explique como se saca la integral de este:

Answer 1

The integral distributes over the sum:

`\displaystyle \int \left(4x^(\frac34) + 2x^(\frac34) + 4x^(\frac12) + 2x + 2\right) \, dx \\\\\\ = \int 4x^(\frac34) \, dx + \int 2x^(\frac34) \, dx + \int 4x^(\frac12) \, dx + \int 2x \, dx + \int 2 \, dx`

Then just integrate each term using the power rule; if n ≠ -1, then

`\displaystyle \int x^n \, dx = (x^(n+1))/(n+1) + C`

Then your integral is simply

`\displaystyle \int \left(4x^(\frac34) + 2x^(\frac34) + 4x^(\frac12) + 2x + 2\right) \, dx \\\\\\ = (4x^(\frac34+1))/(\frac34+1) + (2x^(\frac34+1))/(\frac34+1) + (4x^(\frac12+1))/(\frac12+1) + \frac{2x^2}2 + \frac{2x^1}1 + C \\\\\\ = \frac{16}7 x^(\frac74) + \frac87 x^(\frac74) + \frac83 x^(\frac32) + x^2 + 2x + C \\\\\\ = \boxed{\frac{24}7 x^(\frac74) + \frac83 x^(\frac32) + x^2 + 2x + C}`