Question

Ayo some help over here please :D ! find the indefinite integral ∫1/3x+2 dx

Answer 1

\[\int\frac{1}{3x+2}dx?\]
Have you tried a substitution?
Let u = 3x + 2.

Answer 2

im i subing 3x+2 for X?

Answer 3

\[\text{Letting $u=3x+2$, you get}\\
du=3\;dx\\
\frac{1}{3}du=dx\\
\int\frac{1}{u}\left(\frac{1}{3}du\right)\\
\frac{1}{3}\int\frac{1}{u}du\]
That's the goal of a substitution like this: to get the original integral into a form that's more familiar.

Answer 4

so you subtracted 3x+2 to get 1 and put 3 on the bottom ?

Answer 5

No.
By letting
u = 3x + 2 ... ... ... ... (1),
I'm allowing myself to replace the 3x + 2 in the denominator with the simpler term u. Then, finding the differentials of both sides (implicit differentiation), you get
du = 3 dx.
You have to find a proper sub for dx, since you're changing an integral with respect to x to an integral with respect to u.
However, the given integral only has a dx, so I have to divide both sides of the most recent equation by 3, giving me
1/3 du = dx ... ... ... .. (2).
Now, I substitute everything I can using (1) and (2).

Answer 6

ohhhh okay lol thanks ! i kinda get them better now

Answer 7

we can deduce from the above solution, that whenever you have the derivative of the function in the numerator at the denominator you end up with a natural log, by induction whenever you have the derivative of a function, then you may use the elementary methods and functionality to solve any integral of that case.

Answer 8

@mathsmind, of course when you make that conclusion, you should still make room for sign changes. For example, if the denominator is of the form (1-x). But the idea stays the same.

Answer 9

yes including factors but i believe students need to visualize those equations in its general term, because from my experience they repeat the same mistakes over and over ... so when we teach them general lema the probability of errors are reduced...