[tex] \star) \int \frac{4}{ \sqrt{ {x}^{2} - 16} } + \frac{8}{ \sqrt{ {x}^{2} + 16} } \: dx[/tex]
[tex] = \int \frac{4}{ \sqrt{ {x}^{2} - 16} } \: dx + \int \frac{8}{ \sqrt{ {x}^{2} + 16} } \: dx[/tex]
[tex] = 4 \int \frac{1}{ \sqrt{ {x}^{2} - 16 } } \: dx + 8 \int \frac{1}{ \sqrt{ {x}^{2} + 16} } \: dx[/tex]
[tex] = 4ln(x + \sqrt{ {x}^{2} - 16} ) + 8ln(x + \sqrt{ {x}^{2} + 16} ) + C[/tex]
[tex] \star) \int \frac{6}{ {x}^{2} - 9} + \frac{4}{ \sqrt{ {x}^{2} + 25 } } \: dx[/tex]
[tex] = \int \frac{6}{ {x}^{2} - 9} \: dx + \int \frac{4}{ \sqrt{ {x}^{2} + 25 } } \: dx[/tex]
[tex] = 6 \int \frac{1}{ {x}^{2} - {3}^{2} } \: dx + 4 \int \frac{1}{ \sqrt{ {x}^{2} + 25} } \: dx[/tex]
[tex] = 6 \times \frac{1}{2 \times 3} ln | \frac{x - 3}{x + 3} | + 4ln(x + \sqrt{ {x}^{2} + 25} ) + C[/tex]
[tex] = ln | \frac{x - 3}{x + 3} | + 4ln(x + \sqrt{ {x}^{2} + 25 } ) + C[/tex]
