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I was following through on symbolab but they did a step which completely threw me off Solution provided by Symbolab

my biggest problem is finding out how did they get 5u

Integral by substitution

Full solution can be found here

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    Sometimes, looking at 1-step substitutions can be a little bit hard to wrap your head around. In that case, just go through a few other steps. It only takes 5 extra seconds:) – 2017-02-23

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Well, they did this because $25=5^2$. We want $x=5u$, because that would make the function $\frac{75}{x^2+25}$ into a function which is easier to integrate, namely$$\frac{C}{u^2+1}$$ Where $C$ is a constant. You may wish to note that integrating $$\frac{1}{1+u^2}$$ is well known; it is $\arctan(u)$.

So we're substituing $x=5u$ in order to make the integral into a integral that is well known.

Technically speaking, you could just say that you're substituing $x=5 \tan u$, assuming the fact that you do not know this "well known integral". It would still work.

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    Understood that but how did he factor out 25 without affecting the x – 2017-02-23
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    @user2861799 Could you clarify? Just putting in $x=5u$, gives us, as mentioned in my answer $$\frac{C}{u^2+1}$$ and since $\mathrm{d}x=5 \mathrm{d}u$, we have that $C=15$ in this case. The integral of $$\frac{C}{u^2+1}$$ is $C \arctan u$. – 2017-02-23
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    $x = \frac{\sqrt a}{\sqrt b}$ therefore shouldnt $x = \frac{1}{5}u$ – 2017-02-23
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    @user2861799 $b=1, a=25$ in this case. – 2017-02-23
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    OH okay lol i just got it 12: am... sleepy and at work lol thanks – 2017-02-23
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$\int \frac{75}{x^2+25} \,dx = \int \frac{3}{\left(\frac{x}{5}\right)^2+1} \,dx = 15\int\frac{\frac{1}{5}}{\left(\frac{x}{5}\right)^2+1}\,dx$.

Apply a u-subsitution with u = x/5 and du=dx/5

Then you get 15arctan(x/5)+C

i gave up on the latex lol