Question

Ma ajutați va rog la mate clasa a 12 a cele încercuite doar

Answer 1

$k) \int \frac{1}{ \sqrt{6 {x}^{2} + 24 } } \: dx$

$= \int \frac{1}{ \sqrt{6( {x}^{2} + 4)} } \: dx = \frac{1}{ \sqrt{6} } \int \frac{1}{ \sqrt{ {x}^{2} + 4} } \: dx$

$= \frac{ \sqrt{6} }{6} \times ln(x + \sqrt{ {x}^{2} + 4} ) + C$

$l) \int \frac{1}{ \sqrt{2 {x}^{2} - 18} } \: dx$

$= \int \frac{1}{ \sqrt{2( {x}^{2} - 9)} } \: dx = \frac{1}{ \sqrt{2} } \int \frac{1}{ {x}^{2} - 9 } \: dx$

$= \frac{ \sqrt{2} }{2} \times ln( x + \sqrt{ {x}^{2} - 9} ) + C$

$m) \int \frac{ \sqrt{3} }{ \sqrt{48 - 3 {x}^{2} } } \: dx$

$= \int \frac{ \sqrt{3} }{ \sqrt{3(16 - {x}^{2} )} } \: dx = \frac{ \sqrt{3} }{ \sqrt{3} } \int \frac{1}{ \sqrt{16 - {x}^{2} } } \: dx$

$= \int \frac{1}{ \sqrt{ {4}^{2} - {x}^{2} } } \: dx = arcsin \: \frac{x}{4} + C$

$h) \int \frac{ \sqrt{ {x}^{2} + 4 } - 1}{ {x}^{2} + 4} \: dx$

$= \int \frac{ \sqrt{ {x}^{2} + 4 } }{ {x}^{2} + 4 } \: dx - \int \frac{1}{ {x}^{2} + 4} \: dx$

$= \int \frac{1}{ \sqrt{ {x}^{2} + 4} } \: dx - \int \frac{1}{ {x}^{2} + 4} \: dx$

$= \int \frac{1}{ \sqrt{ {x}^{2} + {2}^{2} } } \: dx - \int \frac{1}{ {x}^{2} + {2}^{2} } \: dx$

$= ln(x + \sqrt{ {x}^{2} + 4} ) - \frac{1}{2} arctg \frac{x}{2} + C$

$i) \int \frac{ \sqrt{ {x}^{2} - 4} + 4}{ {x}^{2} - 4} \: dx$

$= \int \frac{ \sqrt{ {x}^{2} - 4 } }{ {x}^{2} - 4} \: dx + \int \frac{4}{ {x}^{2} - 4} \: dx$

$= \int \frac{1}{ \sqrt{ {x}^{2} - 4} } \: dx + 4 \int \frac{1}{ {x}^{2} - 4} \: dx$

$= ln(x + \sqrt{ {x}^{2} - 4 } ) + 4 \times \frac{1}{4} \times ln \frac{ |x - 2| }{ |x + 2| }$

$= ln(x + \sqrt{ {x}^{2} - 4} ) + ln \frac{ |x - 2| }{ |x + 2| } + C$

$j) \int \frac{ \sqrt{2 - {x}^{2} } + \sqrt{ {x}^{2} + 2} }{ \sqrt{4 - {x}^{4} } } \: dx$

$= \int \frac{ \sqrt{2 - {x}^{2} } }{ \sqrt{4 - {x}^{4} } } \: dx + \int \frac{ \sqrt{ {x}^{2} + 2 } }{ \sqrt{4 - {x}^{4} } } \: dx$

$= \int \frac{ \sqrt{2 - {x}^{2} } }{ \sqrt{(2 - {x}^{2})(2 + {x}^{2} )} } \: dx + \int \frac{ \sqrt{ {x}^{2} + 2 } }{ \sqrt{(2 - {x}^{2} )(2 + {x}^{2} )} } \: dx$

$= \int \frac{1}{ \sqrt{2 + {x}^{2} } } \: dx + \int \frac{1}{ \sqrt{2 - {x}^{2} } } \: dx$

$= ln(x + \sqrt{2 + {x}^{2} } ) + arcsin \: \frac{x}{ \sqrt{2} }$

$= ln(x + \sqrt{2 + {x}^{2} } ) + arcsin \frac{ \sqrt{2} x}{2} + C$

$k) \int \frac{2x + 1}{ \sqrt{ {x}^{2} - 16 } } \: dx$

$= \int \frac{2x}{ \sqrt{ {x}^{2} - 16} } \: dx + \int \frac{1}{ \sqrt{ {x}^{2} - 16} } \: dx$

$= 2 \int \frac{x}{ \sqrt{ {x}^{2} - 16 } } \: dx + \int \frac{1}{ \sqrt{ {x}^{2} - 16 } } \: dx$

$= 2 \sqrt{ {x}^{2} - 16} + ln(x + \sqrt{ {x}^{2} - 16} ) + C$