Question

$\int x^2\cdot \cos \left(x^3\right)\sqrt{{\sin }^7\left(x^3\right)}\cdot dx$

Answer 1

$Letsin\left(x^3\right)=tthencos{x}^3\times 3x^{2}dx=dtor{x}^{2}cos\left(x^3\right)dx=\left(\frac{dt}{3}\right)\therefore I=\left(\frac{1}{3}\right)\int \sqrt{t^7}dt=\left(\frac{1}{3}\right)\int t^{\left(\frac{7}{2}\right)}dt=\left(\frac{1}{3}\right)\times \left(\frac{t^{\left(\frac{9}{2}\right)}}{\left(\frac{9}{2}\right)}\right)+C=\left(\frac{2}{27}\right)\sqrt[9]{9sin\left(x^3\right)}+C$