1 - 3 Examples | Indefinite Integrals

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

$\,\,\,\,\,\,\,\,\,\, = \displaystyle \int \left( \dfrac{2x^3}{x^2} + \dfrac{5x^2}{x^2} - \dfrac{4}{x^2} \right) dx$

$\,\,\,\,\,\,\,\,\,\, = \displaystyle \int ( 2x + 5 - 4x^{-2} ) \, dx$

$\,\,\,\,\,\,\,\,\,\, = \displaystyle \int ( 2x + 5 - 4x^{-2} ) \, dx$

$\,\,\,\,\,\,\,\,\,\, = \displaystyle 2\int x\,dx + 5\int dx - 4\int x^{-2} \, dx$

$\,\,\,\,\,\,\,\,\,\, = \dfrac{2x^2}{2} + 5x - \dfrac{4x^{-1}}{-1} + C$

$\,\,\,\,\,\,\,\,\,\, = x^2 + 5x + \dfrac{4}{x} + C$          answer

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

This is the form $\displaystyle \int u^n \, du$. If we let $u = x^4 - 5x^2 - 6x$, then $u$ is raised to a power 4 and is multiplied by the differential $du = (4x^3 - 10x - 6) \, dx$ of the function corresponding to $u$, the integral can be evaluated as follows:
$\displaystyle \int (x^4 - 5x^2 - 6x)^4 (4x^3 - 10x - 6) \, dx = \dfrac{(x^4 - 5x^2 - 6x)^5}{5} + C$          answer
 

It should be pointed out that no integral can be evaluated directly unless it contains, in addition to the expression identified with $u^n$, the exact differential of the function corresponding to $u$.

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

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

$\,\,\,\,\,\,\,\,\,\, = \dfrac{y^{3/2}}{3/2} + \dfrac{y^{5/2}}{5/2} + C$

$\,\,\,\,\,\,\,\,\,\, = \frac{2}{3}\,y^{3/2} \,+\, \frac{2}{5}\,y^{5/2} + C$           answer