Question

Evaluate,
m) sinºx dx​

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

AnswEr :

Given Expression,

$\displaystyle \sf l = \int {sin}^{3} x.dx \\ \\ \longrightarrow\displaystyle \sf l = \int ( sin \: x. {sin}^{2} x)dx$

Since, sin²x = 1 - cos²x

$\longrightarrow \: \displaystyle \sf l = \int {sin}^{} x(1 - {cos}^{2}x) .dx$

Let t = cos(x)

Differentiating w.r.t x on both sides,

$\implies \sf \dfrac{dt}{dx}= - sin \: x \\ \\ \implies \sf \: dx = - \dfrac{dt}{sin \: x}$

Therefore,

$\longrightarrow \: \displaystyle \sf l = -\int { \cancel{sin}}^{} x(1 - {t}^{2}) . \dfrac{dt}{ \cancel{sin \: x} } \\ \\ \longrightarrow \displaystyle \sf l = \int {t}^{2} dt - \int dt \\ \\ \longrightarrow \sf l = \dfrac{ {t}^{3} }{3} - t + c \\ \\ \longrightarrow \boxed{ \boxed{ \sf l = \dfrac{ {cos}^{3} x}{3} - cos \: x + c}}$