how to evaluate the integral

Question

darkigloo · Answer

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

anonymous · Answer

Have you learned substitution?

anonymous · Answer

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

darkigloo · Answer

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

anonymous · Answer

Ok, so how come you didn't finish?

darkigloo · Answer

i dont know what to do from there.

darkigloo · Answer

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

anonymous · Answer

Okay, so what do you have so far?

anonymous · Answer

What do you get after your $u$ sub?

darkigloo · Answer

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

anonymous · Answer

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

darkigloo · Answer

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

anonymous · Answer

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

anonymous · Answer

Remember that $u$ and $x$ are dummy variables.

anonymous · Answer

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

darkigloo · Answer

=2(-7)=-14?

anonymous · Answer

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

anonymous · Answer

Because $x~dx = \frac 12 du$.

darkigloo · Answer

oh ok. thank you.