Question

Lets say i know the substitution method, but i cant use it on a test, how do i find the antiderivative of a function that involves the chain rule?

Answer 1

give an example!

Answer 2

wait

Answer 3

\[\int\limits_{}^{} ((5x+8)^.5+11)/\sqrt5x+8)\]

Answer 4

the .5 obviously represent raised to 1/2, and the whole term 5x+8 to the right of the dividion sign is under the square root

Answer 5

u can use substitution here, substitue 5x+8 power half as something, then its derivative will have 1/root of(5x+8).. then you can easily integrate

Answer 6

sorry for the late reply

Answer 7

Yeah, he is saying that the class hasn't done it yet, and he's concerned he will be 'marked down' for using substitution to solve integrals that involve the chain rule.

However there really is no other way to solve them, so either you will not be given integrals which involve the chain rule, or you will be allowed to use substitution.

Answer 8

but cant we divide

Answer 9

In that case you could do the division yes:
\[\int \frac{\sqrt{5x + 8} + 11}{\sqrt{5x+8}}dx = \int 1 + \frac{11}{\sqrt{5x+8}}dx\]

But how will you solve the rightmost term without chain rule?

Answer 10

you can.. lol its direct integral.. sorry i thought you wanted to use substitution!!

Answer 11

right side is of the the form 1/2rootx whos integral is root x

Answer 12

well, i think we can factor out the 11, and be left with sqrt5x+8

Answer 13

then intergrate from there, giving us x+22(5x+8)^1/2

Answer 14

Nerp.

Answer 15

the second term will have 5 in the denominator!

Answer 16

Still need the chain rule because you'll be missing a factor of 5.

Answer 17

we can divide by 5

Answer 18

oh, wait i am still getting a 5 in the denominator

Answer 19

Err blah

Answer 20

where are you getting a 5 in the denominator if it's not from the chain rule?

Answer 21

after i integrate i get x+22(5x+8)^1/2

Answer 22

, so now i want ot check that this is right so i take the derivative

Answer 23

how should i ajust for the cahin rule

Answer 24

If 22(5x+8)^1/2 is the integral of 11/sqrt(5x+8) then the derivative of 22(5x+8)^1/2 should be 11/sqrt(5x+8).

Answer 25

i am not sure, did you figure out the integral

Answer 26

Yes. But you have to use a substitution.

Answer 27

no way to do it without

Answer 28

\[Let\ u = 5x+8\]\[\implies du = 5dx\]\[\implies dx = \frac{1}{5}du\]

\[11\int \frac{1}{\sqrt{5x+8}}dx = 11\int\frac{1}{\sqrt{u}}(\frac{1}{5})du\]

Answer 29

Nope, you cannot avoid it just like you cannot avoid the chain rule when you take the derivative.

Answer 30

man, i remeber my prof, doin git in classs witout substitution, i got o go look at my notes

Answer 31

Nope. If you try you will be missing the factor of 1/5.