Integration!! Pls Help!!!

[Only1Me]

the integral of (dx) / x^2 (sqrt x^2-1)

table on integrals questions:

the integral of x tan^-1 dx

the intgral of sin 2x cos 3x dx

 

[tkhunny]

\(\displaystyle \sqrt{1-\frac{1}{x^{2}}}+C\)

Ill-formed, but I'll guess

\(\displaystyle atan(x)\left(\frac{x^{2}}{2}+\frac{1}{2}\right)-\frac{x}{2} + C\)

\(\displaystyle \frac{\cos(x)}{2} - \frac{cos(5x)}{10}+C\)

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[galactus]

the integral of (dx) / x^2 (sqrt x^2-1)

\(\displaystyle \int\frac{1}{x^{2}\sqrt{x^{2}-1}}dx\)

If we let \(\displaystyle x=sec({\theta}), \;\ dx=sec({\theta})tan({\theta})d{\theta}\)

Then make the subs and it whittles down to:

\(\displaystyle \int cos({\theta})d{\theta}\)

the integral of x tan^-1 dx

\(\displaystyle \int xtan^{-1}(x)dx\)

Use parts. Let \(\displaystyle u=tan^{-1}(x), \;\ dv=xdx, \;\ v=\frac{x^{2}}{2}, \;\ du=\frac{1}{1+x^{2}}dx\)

the intgral of sin 2x cos 3x dx

\(\displaystyle \int sin(2x)cos(3x)dx\)

Rewrite as \(\displaystyle \frac{1}{2}\int sin(5x)dx-\frac{1}{2}\int sin(x)dx\)