Question

∫ sin³x cos⁴x dx
evaluate..​

Answer 1

$\longrightarrow \displaystyle \int \sf{sin}^{3} x \: \: {cos}^{4} x \: dx \\ \\ \longrightarrow \displaystyle \int \sf{sin}^{2} x \: \:sin \:x \: \: {cos}^{4} x \: \: dx\\ \\ \longrightarrow \displaystyle \int \sf(1 - {cos}^{2}x) \: \: {cos}^{4} x \: \: sin \:x \: \: dx \\ \\ \\ \sf let \: \: cos \: x \: = t \\ \sf- sin \: x \: dx \: = dt \\ \boxed{ \sf sin \: x \: dx \: = - dt} \\ \\ \longrightarrow \displaystyle - \int \sf(1 - {t}^{2}) \: {t}^{4} \: dt \\ \\ \longrightarrow \displaystyle - \int \sf ({t}^{4} - {t}^{6} ) \: \: dt \\ \\ \longrightarrow \displaystyle - \int \sf {t}^{4} dt + \displaystyle \int \sf{t}^{6} \: \: dt \\ \\ \longrightarrow \displaystyle - \sf \frac{{t}^{5}}{5} + \sf \frac{{t}^{7} }{7} + c \\ \\ \sf now, \: put \: t = cos \: x \\ \\ \longrightarrow \displaystyle - \sf \frac{{cos}^{5}x}{5} + \sf \frac{{cos}^{7} x}{7} + c$