Question

$\int {\sin }^{4}x{\cos }^{3}xdx$

• A. $\frac{5{\sin }^{5}x-7{\sin }^{7}x}{35}+C$
• B. $\frac{7{\sin }^{5}x-5{\sin }^{7}x}{35}+C$
• C. $\frac{7{\sin }^{5}x-3{\sin }^{7}x}{21}+C$
• D. $\frac{5{\sin }^{5}x+5{\sin }^{7}x}{25}+C$

Answer 1

Answer: (B)

$\frac{7{\sin }^{5}x-5{\sin }^{7}x}{35}+C$

$\int {\sin }^{4}x{\cos }^{3}x\text{dx}$

${\cos }^2=1-{\sin }^{2}x$

$\int \cos x\left(-{\sin }^{4}x\right)\left({\sin }^{2}x-1\right)dx$

$u=\sin x$

$dx=\frac{1}{\cos \left(x\right)}dx$

$-\int u^4\left(u^2-1\right)du$

$-\int u^6-u^{4}du$

$-\int 4^{6}du+\int 4^{4}du$

$\frac{-4}{7}+\frac{4^5}{5}+c$

$\frac{{\sin }^{5}x}{5}-\frac{{\sin }^{7}x}{7}+c$

$\frac{7{\sin }^{5}x-5{\sin }^{7}x}{35}+c$