Mathematics

Evaluate $$\int { \sin ^{ \tfrac{3}{4} }{ x } } \cos { x } dx$$.


SOLUTION
$$\displaystyle\int{{\sin}^{\tfrac{3}{4}}{x}\cos{x}dx}$$

Put $$t=\sin{x}\Rightarrow dt=\cos{x}dx$$

$$=\displaystyle\int{{t}^{\tfrac{3}{4}}dt}$$

$$=\dfrac{{t}^{\tfrac{3}{4}+1}}{\dfrac{3}{4}+1}+c$$

$$=\dfrac{{t}^{\tfrac{3+4}{4}}}{\dfrac{3+4}{4}}+c$$

$$=\dfrac{{t}^{\tfrac{7}{4}}}{\dfrac{7}{4}}+c$$

$$=\dfrac{4}{7}{t}^{\tfrac{7}{4}}+c$$ , where $$c$$ is the constant of integration

$$=\dfrac{4}{7}{\sin}^{\tfrac{7}{4}}{x}+c$$, where $$t=\sin{x}$$
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