How do you use integration by parts to establish the reduction formula $\int {cos}^{n} (x) d x = (\frac{1}{n}) {cos}^{n - 1} (x) sin (x) + \frac{n - 1}{n} \int {cos}^{n - 2} (x) d x$ $intcos^n(x) dx = (1/n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx$ ?

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How do you use integration by parts to establish the reduction formula $\int {cos}^{n} (x) d x = (\frac{1}{n}) {cos}^{n - 1} (x) sin (x) + \frac{n - 1}{n} \int {cos}^{n - 2} (x) d x$ $intcos^n(x) dx = (1/n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx$ ?

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Answer 1 · 2014-09-11T18:29:02

Let
$I = \int {cos}^{n} x d x$ $I=int cos^nxdx$ .

Let $u = {cos}^{n - 1} x$ $u=cos^{n-1}x$ and $d v = cos x d x$ $dv=cosxdx$
$\Rightarrow d u = - (n - 1) {cos}^{n - 2} x sin x d x$ $Rightarrow du=-(n-1)cos^{n-2}x sinxdx$ and $v = sin x$ $v=sinx$

By Integration by Parts,
$I = {cos}^{n - 1} x sin x + (n - 1) \int {cos}^{n - 2} x {sin}^{2} x d x$ $I=cos^{n-1}x sinx+(n-1)int cos^{n-2}xsin^2xdx$
By ${sin}^{2} x = 1 - {cos}^{2} x$ $sin^2x=1-cos^2x$ ,
$I = {cos}^{n - 1} x sin x + (n - 1) \int {cos}^{n - 2} x (1 - {cos}^{2} x) d x$ $I=cos^{n-1}x sinx+(n-1)int cos^{n-2}x(1-cos^2x)dx$
By splitting the last integral,
$I = {cos}^{n - 1} x sin x + (n - 1) \int {cos}^{n - 2} x d x - (n - 1) I$ $I=cos^{n-1}x sinx+(n-1)int cos^{n-2}xdx-(n-1)I$
By adding $(n - 1) I$ $(n-1)I$ ,
$n I = {cos}^{n - 1} x sin x + (n - 1) \int {cos}^{n - 2} x d x$ $nI=cos^{n-1}x sinx+(n-1)int cos^{n-2}xdx$
By dividing by $n$ $n$ ,
$I = \frac{1}{n} {cos}^{n - 1} x sin x + \frac{n - 1}{n} \int {cos}^{n - 2} x d x$ $I=1/ncos^{n-1}xsinx+{n-1}/nintcos^{n-2}xdx$

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How do you use integration by parts to establish the reduction formula $\int {cos}^{n} (x) d x = (\frac{1}{n}) {cos}^{n - 1} (x) sin (x) + \frac{n - 1}{n} \int {cos}^{n - 2} (x) d x$ $intcos^n(x) dx = (1/n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx$ ?

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How do you use integration by parts to establish the reduction formula $\int {cos}^{n} (x) d x = (\frac{1}{n}) {cos}^{n - 1} (x) sin (x) + \frac{n - 1}{n} \int {cos}^{n - 2} (x) d x$ $intcos^n(x) dx = (1/n)cos^(n-1)(x)sin(x)+(n-1)/nintcos^(n-2)(x)dx$ ?