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Evaluate: $\int\limits\cos^{5} x dx$

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  • If $I_n=\int \cos ^n x dx$ then
  • $I_n=\large\frac{1}{n}$$ \cos ^{n-1} x \sin x+\large\frac{n-1}{n}$$ I_n-2$
$I_5=\int \cos ^5 x dx$
Step 1:
$\qquad=\large\frac{1}{5}$$ \cos ^4 x \sin x +\large\frac{5-1}{5} $$I_3$
Step 2:
$\qquad=\large\frac{1}{5}$$ \cos ^4 x \sin x +\large\frac{4}{5} \bigg[\large\frac{1}{3} $$\cos^2 x \sin x+\large\frac{3-1}{3}$$I_1\bigg]$
Step 3:
$\qquad=\large\frac{1}{5}$$ \cos ^4 x \sin x +\large\frac{4}{15} $$\cos^2 x \sin x+\large\frac{8}{15 } \int $$\cos x dx $
Step 4:
$\qquad=\large\frac{1}{5}$$ \cos ^4 x \sin x +\large\frac{4}{15} $$\cos^2 x \sin x+\large\frac{8}{15 } $$\sin x +c$
answered Aug 14, 2013 by meena.p
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