Zadatak 0032

Test: MPsIVKVIII_006

 

Reši integrale: a) \(\int \frac{\left ( \arcsin x \right )^{2}}{\sqrt{1-x^2}}dx\);  b) \(\int x^2\ln xdx\)

 

Rešenje:


 

 

a) \(...=\left ( t= \arcsin x, dt=\frac{dx}{\sqrt{1-x^2}}\right )\)

 

\(=\int t^2dt=\frac{t^3}{3}+C=\frac{1}{3}(\arcsin x)^3+C\)

 

b) \(...=\left ( u=lnx, du =\frac{dx}{x}; v=\int x^2dx =\frac {x^3}{3}\right )\)

 

\(=\frac{x^3}{3}\ln x-\frac{1}{3}\int x^3\cdot \frac{dx}{x}\)

 

\(=\frac{x^3}{3}\ln x-\frac{1}{3}\int x^2dx\) \(=\frac{x^3}{3}\ln x-\frac{x^3}{9}+C\).