This problem will show you how to compute ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{3t}cos(t)dt.$

Let $I={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{3t}cos(t)dt.$ Use integration by parts with $u=e^{3t}$ and $dv=cos(t)dt$ to write a formula for I in terms of another integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{3t}sin(t)dt.$ Then, do integration by parts again on the new integral, this time using $U=e^{3t}$ and $dV=sin(t)dt.$ This should give you a formula in which the original integral I appears again. Putting it all together, you get an equation

$I=($ some stuff $)($ something $)*I$

Write down this equation and solve for I to finish computing the integral. Finally, check your work by taking the derivative of your answer and making sure that you get $e^{3t}cos(t).$

$($If something confuses you about this procedure, look in the textbook at Example $4$ in Section $7\mathrm{.}1,$ which is very similar.$)$

$4.($a$)$ Use integration by parts to compute ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{ln(x)}{x}dx.$ Hint: You should see something similar to Question $3.$

$($b$)$ Use substitution to compute ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{ln(x)}{x}dx.$ Check that you get the same answer as in part $($a$).$