Question

найти интеграл, 33 балла!
$x ^{3} (1 - 5x ^{2} )^{10} : dx$
подробно, пожалуйста.​

Answer 1

$intlimits {x^{3}(1 - 5x^{2})^{10}} , dx=intlimits dfrac{-10xcdot 5x^{2}(1 - 5x^{2})^{10}}{-50} , dx = left|begin{array}{ccc}1 - 5x^{2} = t,\ 5x^{2} = 1 - t, \dt = -10x dx\end{array}right|= \=intlimits {dfrac{(1-t)t^{10}}{-50} } , dx = dfrac{1}{50} intlimits {(t^{11} - t^{10})} , dx=dfrac{1}{50} cdot left(dfrac{t^{12}}{12} -dfrac{t^{11}}{11}right)+C =\=dfrac{t^{12}}{600}-dfrac{t^{11}}{550} +C = dfrac{(1 - 5x^{2})^{12}}{600}-dfrac{(1 - 5x^{2})^{11}}{550} +C$