Question

Va rog sa ma ajutati la rezolvarea a 2 integrale, nu inteleg cum sa le fac:

f(x)= a^x* e^x;

f(x)= 2^x* e^x,

multumesc..

Answer 1

  1) 
 $f(x)= a^{x}*e^{x} \\ f(x)=(a*e)^{x} \\ \int { (a*e)^{x} } \, dx = \frac{ (a*e)^{x} }{ln(a*x)}=\frac{ a^{x}*e^{x} }{ln(a)+ln(e)}=\frac{ a^{x}*e^{x} }{ln(a)+1} \\ Conditii: a >0 ..si.. a \neq 1$  

2)
$f(x)= 2^{x}*e^{x} \\ f(x)=(2*e)^{x} \\ \int { (2*e)^{x} } \, dx = \frac{ (2*e)^{x} }{ln(2*x)}=\frac{ 2^{x}*e^{x} }{ln(2)+ln(e)}=\frac{ 2^{x}*e^{x} }{ln(2)+1}$

Am folosit formulele:
 $1) \\ a^{c} * b^{c} = (ab)^{c} ..din ..proprietatile.. puterilor \\ \\ 2) \\ \int { a^{x} } \, dx = \frac{ a^{x} }{ln(a)}$
Conditii:  a ∈ R*+  \ {1}