Question

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

Answer 1

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