sin-10-cos-d-

Question Number 83266 by 09658867628 last updated on 29/Feb/20

\int \sin^{10} Θ \cos Θ d Θ

Commented by Tony Lin last updated on 29/Feb/20

l e t s i n θ = t, d t = c o s θ d θ

\int t^{10} d t

= \frac{t^{11}}{11} + c

= \frac{{(s i n θ)}^{11}}{11} + c

Answered by Rio Michael last updated on 29/Feb/20

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solution

\int \sin^{10} θ \cos θ d θ

let \sin θ = u \Rightarrow \frac{d u}{d θ} = \cos θ

∴ d θ = \frac{d u}{\cos θ}

\Rightarrow \int \sin^{10} θ \cos θ d θ = \int u^{10} \cos θ \frac{d u}{\cos θ}

= \int u^{10} d u = \frac{u^{11}}{11} + k

\Rightarrow \int \sin^{10} θ \cos θ d θ = \frac{\sin^{11} θ}{11} + k