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Evaluate: $\int\limits\sin^{4} x dx$

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1 Answer

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Toolbox:
  • If $I_n=\int \sin ^n x dx$ then
  • $I_n=\large\frac{-1}{n}$$ \sin ^{n-1} x \cos x+\large\frac{n-1}{n}$$ I_n-2$
$I_4=\int \sin ^4 x dx$
Step 1:
$\qquad=\large\frac{-1}{4}$$ \sin ^3 x \cos x +\large\frac{4-1}{4} $$I_2$
Step 2:
$\qquad=\large\frac{-1}{4}$$ \sin ^3 x \cos x +\large\frac{3}{4} \bigg(\large\frac{-1}{2} $$\sin x \cos x+\large\frac{2-1}{2}$$I_0\bigg)$
Step 3:
$\qquad=\large\frac{-1}{4}$$ \sin ^3 x \cos x -\large\frac{3}{8} $$\sin x \cos x+\large\frac{3}{8 } \int $$dx $
Step 4:
$\qquad=\large\frac{-1}{4}$$ \sin ^3 x \cos x -\large\frac{3}{8} $$\sin x \cos x+\large\frac{3}{8 } $$x+c$

 

answered Aug 14, 2013 by meena.p
 
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