Question

$\int {\sin }^{5/2}x{\cos }^{3}x$ $dx=$

• A. $2{\sin }^{7/2}x(\frac{1}{7}+\frac{1}{11}{\sin }^{2}x)+c$
• B. $2{\sin }^{7/2}x(\frac{1}{7}-\frac{1}{11}{\sin }^{2}x)+c$
• C. $3{\sin }^{7/2}x(\frac{1}{5}-\frac{1}{9}{\sin }^{2}x)+c$
• D. $3{\sin }^{7/2}x(\frac{1}{7}-\frac{1}{11}{\sin }^{2}x)+c$

Answer 1

The correct option is B $2{\sin }^{7/2}x(\frac{1}{7}-\frac{1}{11}{\sin }^{2}x)+c$

$\int {\sin }^{5/2}x{\cos }^{3}xdx=\int {\sin }^{5/2}x(1-{\sin }^{2}x)\cos xdx=\int {\sin }^{5/2}x\cos xdx-\int {\sin }^{9/2}x\cos xdx=\frac{2{\sin }^{7/2}x}{7}-\frac{2{\sin }^{11/2}x}{11}$

$=2{\sin }^{7/2}x(\frac{1}{7}-\frac{1}{11}{\sin }^{2}x)+c$