Question

Evaluate the integral using integration by parts where possible.
(3x − 5)4^(x) dx

Answer 1

\[\Large \bf \int\limits \color{#008353}{(3x-5)}\color{#DD4747}{4^x\;dx}\]
By Parts:\[\Large\bf \color{#008353}{u=(3x-5)},\qquad\qquad\qquad \color{#DD4747}{dv=4^x\;dx}\]

Answer 2

Do you remember how to integrate this term, 4^x? :)

Answer 3

no i couldn't remember! :/

Answer 4

\[\Large (a^x)'\quad=\quad a^x(\ln a)\]When we integrate, we'll end up dividing by this extra factor instead of multiplying.\[\Large \int\limits a^x\;dx \quad=\quad \frac{1}{\ln a}a^x+C\]

Answer 5

\[\Large\bf \color{#008353}{u=(3x-5)},\qquad\qquad\qquad \color{#DD4747}{dv=4^x\;dx}\]\[\Large\bf \color{#008353}{du=3dx},\qquad\qquad\qquad\qquad \color{#DD4747}{v=?}\]So what do you get for your v? :)

Answer 6

so is it (1/ln4)4^(x)??

Answer 7

good good good.

Answer 8

what do i do after that?

Answer 9

\[\Large\bf \color{#008353}{u=(3x-5)},\qquad\qquad\qquad \color{#DD4747}{dv=4^x\;dx}\]\[\Large\bf \color{#008353}{du=3dx},\qquad\qquad\qquad\qquad \color{#DD4747}{v=\frac{1}{\ln4}4^x}\]So we have our parts.

Answer 10

So now we plug them into our formula:
\[\Large\bf \int\limits u\;dv\quad=\quad uv-\int\limits v\;du\]

Answer 11

\[\Large\bf =\quad\color{#008353}{(3x-5)} \color{#DD4747}{\left(\frac{1}{\ln4}4^x\right)}-\int\limits \color{#DD4747}{\left(\frac{1}{\ln4}4^x\right)}\color{#008353}{3dx}\]

Answer 12

something like that, yes? :o
Looks like we have another integral to deal with.
This one, luckily, doesn't require By Parts.

Answer 13

okay so i get a little confused with the ln integral

Answer 14

There in no variable inside of the log, so just ignore it.
We can pass it out of the integral since it's a constant.\[\Large \int\limits\left(\frac{1}{\ln4}4^x\right)3dx\quad=\quad \frac{3}{\ln4}\int\limits 4^x\;dx\]

Answer 15

ohhhhhhh i think i got it :)

Answer 16

(3/ln4)(1/1n4)?

Answer 17

for our second integral there?
\[\Large \frac{3}{\ln4}\cdot\color{#A57F02 }{\int\limits\limits 4^x\;dx}\quad=\quad \frac{3}{\ln4}\cdot\color{#A57F02 }{\frac{1}{\ln4}4^x}\]

Answer 18

so (3/ln^(2)4)4^(x)

Answer 19

For the second integral, yes.
Don't forget about the rest of the problem though :D

Answer 20

ahhh got it! thank you :)

Answer 21

cool c: