Integrales Exponenciales y Logaritmicas

Derivative of an exponential base a:

 $\left(a^u\right)'=?$  

•  $a^u\cdot\ln\left(a\right)\cdot u'$  

•  $a^u\cdot u'$  

•  $a^u\cdot\ln\left(a\right)$  

•  $a^{u'}\cdot\ln\left(a\right)$  

 $\int3x\cdot2^{x^2}dx=?$  

•  $\frac{3\cdot2^{x^2}}{2\ln\left(2\right)}+c$  

•  $\frac{3\cdot2^{x^2}}{2}+c$  

•  $\frac{\ln\left(2\right)\cdot2^{x^2}}{6}+c$  

•  $\frac{6\cdot2^{x^2}}{\ln\left(2\right)}+c$  

• Understood

• ...I'll go on

Derivative of an exponential base e:

 $\left(e^u\right)'=?$  

•  $e^u\cdot u'$  

•  $e^u\cdot\ln\left(u\right)\cdot u'$  

•  $e^{u'}\cdot u$  

•  $e^{u'}\cdot\ln\left(u\right)$  

 $\int e^{7x}dx=?$  

•  $\frac{e^{7x}}{7}+c$  

•  $e^{7x}+c$  

•  $7e^{7x}+c$  

•  $e^{7x}\cdot\ln\left(7\right)+c$  

• Understood

• ...I'll go on

Derivative of a logarithm base a:

 $\left(\log_a\left(u\right)\right)'=?$  

•  $\frac{u'}{u\cdot\ln\left(a\right)}$  

•  $\frac{u'}{u}$  

•  $\frac{1}{u\cdot\ln\left(a\right)}$  

•  $\frac{u'\cdot\ln\left(a\right)}{u}$  

 $\int\frac{20x^4}{2x^5\ln\left(3\right)}dx=?$  

•  $2\log_3\left(2x^5\right)+c$  

•  $2\ln\left(2x^5\right)+c$  

•  $\frac{\log_3\left(2x^5\right)}{2}+c$  

•  $\frac{\ln\left(2x^5\right)}{2}+c$  

• Understood

• ...I'll go on

Derivative of a natural logarithm):

 $\left(\ln\left(u\right)\right)'=?$  

•  $\frac{u'}{u}$  

•  $\frac{1}{u}$  

•  $\frac{1}{u'}$  

•  $\frac{u}{u'}$