Question

$\int {\sin }^{\frac{5}{2}}x{\cos }^{3}xdx=?$

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

Answer 1

The correct option is A $2{\sin }^{\frac{7}{2}}x[\frac{1}{7}+\frac{1}{11}{\sin }^{2}x]+c$

$\begin{array}{c}\int \sin x^{5/2}{\cos }^{3}xdx\\ \Rightarrow \int \sin x^{5/2}\left(1-{\sin }^{2}x\right)\cdot \cos xdx\\ let\sin x=1\\ differentiatew.r.tox\\ \cos xdx=dt\\ dx=\frac{dt}{\cos x}\\ \Rightarrow \int t^{5/2}\left(1-t^2\right)\cdot \cos x\frac{dt}{\cos x}\\ \Rightarrow \int t^{5/2}dt-\int t^{9/2}dt\\ \Rightarrow \frac{t^{7/2}}{7/2}-\frac{t^{11/2}}{11/2}+c\\ \Rightarrow \frac{2}{7}\sin x^{7/2}x\frac{-2}{11}\sin x^{11/2}+c\\ \Rightarrow 2\left[\frac{\sin x^{7/2}}{7}-\frac{\sin x^{11/2}}{11}\right]+c\\ \Rightarrow 2\sin x^{7/2}\left[\frac{1}{7}-\frac{\sin x^2}{11}\right]+cAns.\end{array}$