Question

$\int \sin \left(11x\right)\cdot {\sin }^{9}xdx=$ ______ $+c$.

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

Answer 1

Answer: (A)

$\frac{\sin \left(10x\right)\cdot {\sin }^{10}x}{10}$

Consider, $I=\int \sin \left(11x\right){\sin }^{9}xdx$

$=\int \sin (10x+x){\sin }^{9}xdx$

$=\int \sin 10x\cos x{\sin }^{9}xdx+\int \cos 10x{\sin }^{10}xdx$

$=\frac{{\sin }^{10}x}{10}\sin 10x-\frac{10}{10}\int {\sin }^{10}x\cos 10xdx+\int \cos \left(10x\right){\sin }^{10}xdx$ ..... [Using integration by parts]

$=\frac{{\sin }^{10}x}{10}\sin \left(10x\right)+c$.