All right here's another integral that we can do with u substitution
i really like to emphasize the informal solution of this so i see some function raised to the 20th
if i could get the derivative of that function right next to it then i'm ready to just guess
what the chain rule backwards is doing so i'm i'm going to do this informally first
the derivative of that interior function is just 3. so i'm going to rewrite the integral this way
3x plus 2 to the 20th times 3 dx and i compensate for that factor of 3 by dividing by 3 out in front
and now i'm ready to integrate this just like something raised
to the 20th which gives me that thing to the 21st
divided by 21
and i suppose i should clean it up a little bit i get 1 over 63
times 3x plus 2 to the 21st
plus c so again to build your intuition for this sort of differentiating backwards and considering
the chain rule at the same time it's probably a good idea to differentiate this and make sure that
you see how that reverse process goes so i have d dx
i'm just going to use this form of it 1 3
3x plus 2 to the 21st over 21. and when i apply the derivative the 21 comes down in front
and i subtract 1 from this exponent
and then the chain rule says you better multiply by 3 the derivative of the interior function
and i get what i started with so i know i have the correct anti-derivative
we could also approach this problem formally
i'm just trying to discourage it when it's not necessary so i have let u equal
3x plus 2 in other words i'm trying to view this as a single thing
then du is going to be 3dx
and the way i'll handle it this time is similar to what i did over here
i'm going to force that 3 to happen by just putting it in the integrand and then
i'll compensate for it out in front so i have the integral of 3x plus 2
to the 20th dx
is equal to the integral of 3x plus 2 to the 20th
times 3dx i'm just forcing this du to happen with a 1 3 out in front
and then i get one third
integral well that's just u that was the whole point so u to the 20th and this is just d u
and now i go okay it's a simple power function so i get
u to the 21st over 21. or 1 over 63.
i guess i'll go ahead and plug in the definition of u here real quick so i'm not done until i
write this back in terms of x so i'm going to go back here 1 over 63 times the quantity 3x plus 2
to the 21st
plus c so you can see we get the same answer
i'm just encouraging you to go for the informal solution whenever it's possible