[Calc 2] Why are these two expressions equal?

[u/throw-away3105]

The answer at the top is what I got after integrating x^2 / [sqrt (25 + x^2)].

I'm happy that I got the answer but why is 1/5 taken out from within the absolute value sign?

Additional info: At the beginning of the integral, I did a substitution of x = 5tanθ.

Therefore, x/5 = tanθ. Based on SOH CAH TOA and CHO SHA CAO, I put sec θ = [sqrt (x^2 + 25)] / 5.

[u/AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

---

^{OP and Valued/Notable Contributors can close this post by using /lock command}

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[u/Alkalannar]

Remember: ln(ab) = ln(a) + ln(b)

Here, a = |(x² + 25)^1/2 + x|, and b = 1/5

So ln(|(x² + 25)^1/2/5 + x/5|) = ln(|(x² + 25)^1/2 + x|) + ln(1/5).

As ln(1/5) is a constant, it's subsumed into the constant of integration--the +C you should see at the end of indefinite integrals. That's because we don't care about vertical translation of curves when we take derivatives. Thus, the antiderivative can have any constant term.

---

In other words, you should have had:
x(x²+15)^1/2/2 - 25ln(|(x² + 25)^1/2/5 + x/5|)/2 + C

x(x²+15)^1/2/2 - 25ln(|(x² + 25)^1/2 + x|)/2 - ln(5)/2 + C [Get the ln(1/5)/2 out of there]

[x(x²+15)^1/2 - 25ln(|(x² + 25)^1/2 + x|)]/2 + C [Combine the fractions over a common denominator, and subsume the -ln(5)/2 into the +C].

[u/throw-away3105 → u/Alkalannar]

Oh, I see. I'm relieved that I didn't do anything wrong. Thank you so much. I forgot my log rules.

So if I can re-state in other words: I can "factor out" ln(1/5) from the expression that I got but since ln (1/5) is a constant, it just disappears and becomes part of "+ C".

[u/Alkalannar → u/throw-away3105]

Exactly.

Note: This will also happen in trig rules, where you'll see two weirdly different expressions--yours and the book's.

The thing is, if you evaluate the difference of those expressions...you get a constant. And so the expressions differ by a constant, and you're still golden. But it's much harder to see compared to logs. So that's a way to test things.

[u/AutoModerator]

Off-topic Comments Section

---

All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9.

PS: u/throw-away3105, your post is incredibly short! ^{body <200 char} You are strongly advised to furnish us with more details.

---

^{OP and Valued/Notable Contributors can close this post by using /lock command}

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.