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Pattern - 32 of Indefinite Integration

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⋆ Pattern – 32

$I_n=\cos^n{x} dx$

$\therefore\ \ I_n=\ \int\cos^{n-1}{x}\times cos\ x dx$

$\therefore\ \ I_n=\cos^{n-1}{x}cos \int x dx - \int \left(\frac{d}{dx}\cos^{n-1}{x}\ cos\int{x\ }dx\right)\ dx$

$= \cos^{n-1}{x}\times \sin{x}+\ \int\left(n-1\right)\cos^{n-2}{x}\sin{x\ }\times\sin{x}\ dx$

$=\ {cos}^{n-1}{x}\times s i n{x}+\left(n-1\right)\ \int{cos}^n{x}\ dx$

$=\ {cos}^{n-1}{x}\ sin\ x\ +\ (n-1)\ I_{n-2}-\left(n-1\right)I_n$

$\therefore (1+n-1) I_n = \cos^{n-1}{x}\sin{x}+\left(n-1\right)\ i_{n-2}$

∴ $I_n=\ \frac{\cos^{n-1}{x}\sin{x}}{n}+\ \frac{n-1}{n}\ I_{n-2}\$ : n ≥ 2

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

Solution. Let, $I_5\ =\ \int\ {cos}^5{x}\ dx$

$\therefore\ \ I_5=\ \frac{{cos}^4{x}sin{x}}{5}+\frac{4}{5}\ I_3$

$=\ \frac{{cos}^4{x}sin{x}}{5}+\frac{4}{5}\ \left[\frac{{cos}^2{x}sin{x}}{3}+\frac{2}{3}\ I_3\right]$

$=\ \frac{{cos}^4{x}sin{x}}{5}+\ \frac{4{cos}^2{x}sin{x}}{15}+\frac{8}{15}\ cos\int{x}\ dx$

$=\ \frac{{cos}^4{x}sin{x}}{5}+\ \frac{4{cos}^2{x}sin{x}}{15}+\frac{8}{15}cos{x}+c$