Question

$\int \sin \left(11x\right){\sin }^{9}xdx=$ _____ $+C$

• A. $\frac{\sin \left(10x\right){\sin }^{10}x}{10}$
• B. $\frac{{\sin }^{11}x}{11}$
• C. $\frac{\sin \left(9x\right){\sin }^{9}x}{9}$
• D. $\frac{\cos \left(10x\right){\cos }^{10}x}{10}$

Answer 1

The correct option is A $\frac{\sin \left(10x\right){\sin }^{10}x}{10}$

$\int \sin \left(11x\right){\sin }^{9}xdx$
$=\int \sin (10x+x){\sin }^{9}xdx$
$=\int \sin \left(10x\right){\sin }^{9}x\cos xdx+\int \cos \left(10x\right){\sin }^{10}xdx$
$=\sin \left(10x\right)\frac{{\sin }^{10}x}{10}-\int 10\cos \left(10x\right)\frac{{\sin }^{10}x}{10}dx+\int \cos \left(10x\right){\sin }^{10}xdx+C$
$=\frac{\sin \left(10x\right){\sin }^{10}x}{10}+C$