I need help integrating this.

[icy basin]

Here,

[icy basin]

I need to integrate the following:

$\int \frac{v}{3u-v} dv$

[shy gust]

∫Inheritanc-e ♦

[icy basin]

$\int \frac{v}{3u-v} dv$ \\
\text{ Here, } \\
$a = 3u-v$ \\
$\frac{da}{dv} = -1$ \rightarrow dv = -da$\\
$\int \frac{v}{a} \cdot (-da$) \\
$ -v \int \frac{1}{a} da$ \\
$-v \cdot ln(a) + c$

but it's wrong. Why?

[shy gust]

∫Inheritanc-e ♦
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[daring cobalt]

Substitute $a = 3u - v$. Then $v = 3u-a \implies dv = -da$. The integral becomes:
$$ -\int \frac{3u-a}{a}, da $$
Break it into two integrals.
$$ -\int \frac{3u}{a}, da + \int \frac{a}{a}, da$$
Integrate.
$$-3u\ln(a) + a + C$$
Substitute $a = 3u-v$.
$$-3u\ln(3u-v) + 3u-v + C$$

[shy gust]

waris

[icy basin]

thanks

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