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Evaluate $I=\int x^{3} \cdot \sqrt{x^{2}+4} d x$

case 2: $\sqrt{a^{2}+x^{2}}$

$a^{2}=4 \rightarrow a=2$

let $x=a \tan \theta$

$d x=a \sec ^{2} \theta d \theta$

$\sqrt{x^{2}+4}=a \sec \theta$

$-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$

let $x=2 \tan \theta$

$d x=2 \sec ^{2} \theta d \theta$

$\sqrt{x^{2}+4}=2 \sec \theta$

$I=\int(2 \tan \theta)^3 \cdot 2 \sec \theta \cdot 2 \sec ^{2} \theta d \theta$

$I=\int 8 \tan ^{3} \theta \cdot 4 \sec ^{3} \theta d \theta$

$t=\sec \theta$

$1=32 \int \tan ^{2} \theta \sec ^{2} \theta \cdot(\tan \theta \cdot \sec a) d \theta$

but $\tan ^{2} \theta=\sec ^{2} \theta-1$

$I=32 \int\left(\sec ^{2} \theta-1\right) \cdot \sec ^{2} \theta(\sec \theta \tan \theta) d \theta$

let $t=\sec \theta \rightarrow d t=\sec \theta \tan \theta d \theta$

$I=32 \int\left(t^{2}-1\right) t^{2}(dt)=32 \int t^{4}-t^{2} d t$

$I=32\left[\frac{t^{5}}{5}-\frac{t^{3}}{3} \right]+{c}=32 t^{3}\left[\frac{t^{2}}{5}-\frac{1}{3}\right]+c$

$I=32 t^{3}\left[\frac{3 t^{2}-5}{15}\right]+c=\frac{32}{15} \,t^3\left[3 t^{2}-5\right)+c$

$\tan \theta=\frac{x}{2}$

but I have $t=\sec x=\frac{\sqrt{x^{2}+4}}{2}$

$I=\frac{32}{15}\left(\frac{\sqrt{x^{2}+4}}{2}\right)^{3}\left[3\left(\frac{\sqrt{x^{2}+4}}{2}\right)^{2}-5\right]+c$

Evaluate $\int \frac{d x}{\sqrt{x^{2}-a^{2}}} \:, a>0$

$\sqrt{x^{2}-a^{2}}$ case 2

$x=a\sec \theta$

$d x=a \sec \theta \tan \theta d \theta$

$\sqrt{x^{2}-a^{2}}=a \tan \theta$

$0< \theta < \frac{\pi}{2} \text { or } \pi < \theta < \frac{3 \pi}{2}$

$\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\int \frac{a \sec \theta \tan \theta d \theta}{a \tan \theta}$

$=\int \sec \theta d \theta$

$=\ln |\sec \theta+\tan \theta|+C$

$x=a \sec \theta \rightarrow \sec \theta=\frac{x}{a}$

$\tan \theta=\frac{\sqrt{x^{2}-a^{2}}}{a}$

$\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\ln \left|\frac{x}{a}+\frac{\sqrt{x^{2}+a^{2}}}{a}\right|+c$

$=\ln \left|\frac{x+\sqrt{x^{2}+a^{2}}}{a}\right|+c$

$\ln \frac{a}{b}=\ln a-\ln b$

$\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\ln \left|x+\sqrt{x^{2}+a^{2}}\right|-\ln |a|+c$

suppose that $-\ln |a|+c=c_{1}$

$\int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\ln \left|x+\sqrt{x^{2}+a^{2}}\right|+c_{1}$