What is the integral of $sin (101 x) {sin}^{99} (x) d x$ $sin(101x)sin^99(x)dx$ ?

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What is the integral of $sin (101 x) {sin}^{99} (x) d x$ $sin(101x)sin^99(x)dx$ ?

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Answer 1 · 2018-01-11T13:25:22

$\int sin (101 x) \cdot {(sin x)}^{99} \cdot d x = \frac{1}{100} sin (100 x) \cdot {(sin x)}^{100} + C$ $int sin(101x)*(sinx)^99*dx=1/100sin(100x)*(sinx)^100+C$

Explanation:

$\int sin (101 x) \cdot {(sin x)}^{99} \cdot d x$ $int sin(101x)*(sinx)^99*dx$

= $\int sin (100 x + x) \cdot {(sin x)}^{99} \cdot d x$ $int sin(100x+x)*(sinx)^99*dx$

= $\int (sin (100 x) \cdot cos x + cos (100 x) \cdot sin x) {(sin x)}^{99} \cdot d x$ $int (sin(100x)*cosx+cos(100x)*sinx)(sinx)^99*dx$

= $\int sin (100 x) \cdot {(sin x)}^{99} \cdot cos x \cdot d x$ $int sin(100x)*(sinx)^99*cosx*dx$ + $\int cos (100 x) \cdot {(sin x)}^{100} \cdot d x$ $int cos(100x)*(sinx)^100*dx$

= $\frac{1}{100} sin (100 x) {(sin x)}^{100}$ $1/100sin(100x)(sinx)^100$ - $\int cos (100 x) \cdot {(sin x)}^{100} \cdot d x$ $int cos(100x)*(sinx)^100*dx$ + $\int cos (100 x) \cdot {(sin x)}^{100} \cdot d x$ $int cos(100x)*(sinx)^100*dx$

= $\frac{1}{100} sin (100 x) \cdot {(sin x)}^{100} + C$ $1/100sin(100x)*(sinx)^100+C$

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What is the integral of $sin (101 x) {sin}^{99} (x) d x$ $sin(101x)sin^99(x)dx$ ?

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Explanation:

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