Question

O valor da integral indefinida ∫ (2t * e^6t) dt ?

a) I = t/9 e^6t - 1/3 e^6t + C

b) I = t/3 e^6t + e^6t + C

c) I = t/6 e^6t - 1/9 e^6t + C

d) I = 1/3 e^6t - t/9 e^6t + C

e) I = t/3 e^6t - 1/18 e^6t + C

Answer 1

Neste caso, vamos integrar por partes ... 

$\int\limits {(2t.e^{6t})} \, dx$


$Dividimos\ em\ duas\ ,\ t\ e \ e^{6t}\\\\u=t\ \ du=1.t^{1-1}\ =1\ =\ dt\\\\v=e^{6t}\ \ dv= \frac{1}{6} e^{6t}$

$Temos\ a\ formula\ :\\\\ \int\limits {u.dv} =u.dv- \int\limits\ dv.du$

$Resolvendo:\\\\ 2\int\limits(2t.e^{6t})\ dt\\\\\\2(t. \frac{1}{6}.e^{6t} - \int\limits\ \frac{1}{6}.e^{6t} \ dt)\\\\\\ 2(\frac{t}{6}. e^{6t}- \frac{1}{6} \int\limits\ e^{6t}\ dx \\\\\\2.( \frac{t}{6} .e^{6t}- \frac{1}{6} . \frac{e^{6t}}{6} )\\\\\\2.( \frac{t}{6} .e^{6t}- \frac{e^{6t}}{36} )\\\\\\ \frac{2t}{6}e^{6t} - \frac{2}{36} e^{6t}\ =\ \boxed{\boxed{ \frac{1}{3}e^{6t} - \frac{1}{18}e^{6t} }}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Letra\ e)\ \ \ \ \ \ ok$