Integral Indefinida

Lic. Elsie Hernández S.

Como $\displaystyle {D_{x}(a^{x})=a^{x}ln\;a}$ entonces:

$\displaystyle {\int a^{x} ln\;a\; dx = a^{x} + C}$ y $\displaystyle {\int a^{x} dx = \frac{a^{x}}{ln\;a} + C}$

Ejemplos:

$\displaystyle {\int 2^{x}dx = \frac{1}{ln\;2}\int 2^{x} ln\;2\; dx = \frac{2x}{ln\;2} + C}$
$\displaystyle {\int x\; 3^{4x^{2}} dx = \frac{1}{8}\int 8x\; 3^{4x^{2}}dx = \frac{3^{4x^{2}}}{8} + C}$
$\displaystyle {\int (2t+1)\;5^{t^{2}+t+4}dt }$

$\displaystyle {= \frac{1}{ln\;5}\int (2t+1)\;t^{(t^{2}+t+4)}\; ln\;t\; dt}$

$\displaystyle {=\frac{1}{ln\;5}\; 5^{(t^{2}+t+4)} + C}$