Question

simplify: $\int {\sin }^{4}x{\cos }^{6}xdx$

Answer 1

$\int si{n}^{4}xco{s}^{6}xdx$

$=\int co{s}^{6}x(si{n}^{2}x{)}^{2}dx$

$=\int co{s}^{6}x(1-co{s}^{2}x{)}^{2}dx$

$=\int co{s}^{6}x+co{s}^{10}x-2co{s}^{8}xdx$

$=\int co{s}^{6}xdx+\int co{s}^{10}xdx+\int (-2)co{s}^{8}xdx$

$=\frac{co{s}^{5}xsinx}{6}+\frac{5\int co{s}^{4}x}{6}+\frac{co{s}^{9}xsinx}{10}+\frac{9}{10}\int co{s}^{8}xdx-2(\frac{co{s}^{7}sinx}{8}+\frac{7}{8}\int co{s}^{6}xdx)$

$=\frac{co{s}^{9}xsinx}{10}-\frac{11co{s}^{7}xsinx}{80}+\frac{co{s}^{5}xsinx}{160}+\frac{co{s}^{3}xsinx}{128}+\frac{3cosxsinx}{256}+\frac{3x}{256}$

simplyfying,

$=\frac{2sin10x+5sin8x-10sin6x-40sin4x+20sin2x+120x+c}{10240}$

$=\frac{5sin8x-40sin4x+32si{n}^{5}2x+120x+c}{10240}$