Question

Integrate v^2(cos(v^3))

Answer 1

\[\int\limits_{0}^{1}v^2\cos(v^3)dv\]

Answer 2

Well you see the derivative of something else right? This should hint you in to u-substitution.

Answer 3

Use u-substitution.

Answer 4

is u = v^2 ?

Answer 5

\[u = v^3\]

Answer 6

I suggest after you integrate this, take the derivative and see the connection between u-substitution and the chain rule. They're like two sides of the same coin.

Answer 7

It'll help so that you can kind of see what to look for in future u-substitution problems.

Answer 8

\[\int\limits_{}^{}\frac{ 1 }{ 3 }3v^2 \cos(v^3) dv\]

Answer 9

if u = v^2
du = 2v dv
1/2 u = v dv
if u sub it in,
1/2 integral (u*cos(u*v))dv
is this right so far?
@S0fw0N what?

Answer 10

@S0fw0N
\[\frac{ 1 }{ 3 } \int\limits \cos(u) du\]

Answer 11

yep

Answer 12

@jennychan12

\[u = v^3\]
because when you take du
\[du = 3v^2dx\]
hence,
\[\frac{ du }{ 3 } = x^2dx\]

Answer 13

remember that you can just factor out constants. Hence, you can substitute du for the dx function, which you clearly already have. The point of u-substitution is to make your integration easier to integrate by using the basic integral rules.

Answer 14

yeah i see. i thought cuz u had a squared term that you couldn't use u = v^3.....

Answer 15

ohhhh i see. @abb0t thanks :)

Answer 16

and it's dv, not dx but no biggie :D

Answer 17

You should get
\[\frac{ 1 }{ 3 } \int\limits \cos(u)du \]
where \[du = v^2dv\] and \[u = v^3\]
cos(u) is something you already know how to integrate easily. therefore, once you integrate, you should get a function in the form
f(u) + C
but remember to replace u with the original u-substitution, which was \[u = v^3\]
to give you a final answer in terms of 'v'
f(v) + C