# What is the integral of sin(101x)sin^99(x)dx?

Jan 11, 2018

$\int \sin \left(101 x\right) \cdot {\left(\sin x\right)}^{99} \cdot \mathrm{dx} = \frac{1}{100} \sin \left(100 x\right) \cdot {\left(\sin x\right)}^{100} + C$

#### Explanation:

$\int \sin \left(101 x\right) \cdot {\left(\sin x\right)}^{99} \cdot \mathrm{dx}$

=$\int \sin \left(100 x + x\right) \cdot {\left(\sin x\right)}^{99} \cdot \mathrm{dx}$

=$\int \left(\sin \left(100 x\right) \cdot \cos x + \cos \left(100 x\right) \cdot \sin x\right) {\left(\sin x\right)}^{99} \cdot \mathrm{dx}$

=$\int \sin \left(100 x\right) \cdot {\left(\sin x\right)}^{99} \cdot \cos x \cdot \mathrm{dx}$+$\int \cos \left(100 x\right) \cdot {\left(\sin x\right)}^{100} \cdot \mathrm{dx}$

=$\frac{1}{100} \sin \left(100 x\right) {\left(\sin x\right)}^{100}$-$\int \cos \left(100 x\right) \cdot {\left(\sin x\right)}^{100} \cdot \mathrm{dx}$+$\int \cos \left(100 x\right) \cdot {\left(\sin x\right)}^{100} \cdot \mathrm{dx}$

=$\frac{1}{100} \sin \left(100 x\right) \cdot {\left(\sin x\right)}^{100} + C$