Question

how to evaluate the integral

Answer 1

if f is continuous and \[\int\limits_{0}^{9} f(x) dx = -7 \] find \[\int\limits_{0}^{3} x f(x ^{2}) dx\]

Answer 2

Have you learned substitution?

Answer 3

\[
d(x^2) = 2x~dx \implies x~dx =\frac 12 d(x^2)
\]

Answer 4

i did this: u=x^2, du/dx= 2x, dx=du/2x

Answer 5

Ok, so how come you didn't finish?

Answer 6

i dont know what to do from there.

Answer 7

i do know that F(9)-F(0)=-7

Answer 8

Okay, so what do you have so far?

Answer 9

What do you get after your \(u\) sub?

Answer 10

\[\int\limits_{0}^{3}x f(x ^{2})dx=\int\limits_{0}^{3}2f(u)du\] i don't think that's right

Answer 11

It's wrong, because you are using the same interval for \(dx\) and \(du\). They have different limits.

Answer 12

\[=\int\limits_{0}^{9}2f(u)du\]

Answer 13

Okay, so shouldn't it be obvious by now?

Answer 14

Remember that \(u\) and \(x\) are dummy variables.

Answer 15

Which means that \[
\int f(u)~du = \int f(x)~dx
\]

Answer 16

=2(-7)=-14?

Answer 17

The only mistake though is that it shouldn't be \(2\) but \(1/2\).

Answer 18

Because \(x~dx = \frac 12 du\).

Answer 19

oh ok. thank you.