Question

$\int\limits { \frac{ x^{2} }{ \sqrt{ x^{6}-25 } } } \, dx ; x\ \textgreater \ 5$
$\int\limits { \frac{ 3^x }{ \sqrt{1- 9^{x} } } } \, dx$ ;
$\int\limits { \frac{ x}{ x^{4}+1 } \, dx$
$\int\limits { \frac{ sinx}{ 9-cos^{2} x } \, dx$ ;
$\int\limits { \frac{ cosx}{ sin^{2} x +4 } \, dx$ ;
$\int\limits { \frac{ cosx}{ sin^{2}x-4 } \, dx$ ;

Answer 1

1) int x²/√(x⁶-25) dx
facem sch de var t=x³ si avem dt=dx³=3x²dx
deci integrala=1/3 int dt/√(t²-5²)= 1/3 ln|t+√(t²-5²)| +C= 1/3 ln |x³+√x⁶-25| +C
2) facem sch de var 3^x=t => dt=d3^x=ln 3 *3^x dx
deci int devine 1/ln3 * int dt/√1-t² = 1/ln3 * arcsin t +C= 1/ln3 * arcsin 3^x +C
3) 1/2 int dx²/(x²)²+1=1//2 arctg x² +C
4)- int d(cos x)/(9-cos²x)= int d(cos x)/(cos²x-3²)=1/2*3 int  (1/cos x+3 -1/cos x-3) d cos x= 1/6 ln |(cos x+3)/(cos x-3)| +C
5) int d sinx/ (sin²x+2²)=1/2 arctg sinx /2 +C
6) int d(sin x)/ (sin²x-2²)= 1/4 ln |(sinx +2)/ (sin x -2)| +C