How do you integrate $x \cdot {cos}^{2} (x)$ $x * cos^2 (x)$ ?

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How do you integrate $x \cdot {cos}^{2} (x)$ $x * cos^2 (x)$ ?

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Answer 1 · 2016-11-07T04:42:17

$\frac{x^{2}}{4} + (\frac{x}{4}) sin 2 x + \frac{cos (2 x)}{8} + c$ $x^2/4+(x/4)sin2x+cos(2x)/8+c$ .where c is integration constant .

Explanation:

$\int (x {cos}^{2} x) d x$ $int (xcos^2x)dx$
$= (\frac{1}{2}) \int (x 2 {cos}^{2} x) d x$ $=(1/2)int(x2cos^2x)dx$
now $\int (x 2 {cos}^{2} x) d x$ $int(x2cos^2x)dx$
= $\int {x (1 + cos 2 x)} d x$ $int{x(1+cos2x)}dx$
= $\int x d x + \int (x cos 2 x) d x$ $intxdx+int(xcos2x)dx$
[integration by parts]
= $\frac{x^{2}}{2} + x (\int (cos 2 x) d x) - \int [(\frac{d x}{d x}) \int (cos 2 x) d x] d x$ $x^2/2+x(int(cos2x)dx)-int[(dx/dx)int(cos2x)dx]dx$
= $\frac{x^{2}}{2} + \frac{x sin (2 x)}{2}$ $x^2/2+(xsin(2x))/2$ $- \int (\frac{sin 2 x}{2}) d x$ $-int((sin2x)/2)dx$
= $\frac{x^{2}}{2} + \frac{x sin (2 x)}{2}$ $x^2/2+(xsin(2x))/2$ $+ \frac{cos 2 x}{4}$ $+(cos2x)/4$ $+ c$ $+c$
hence the value of the integral : $\frac{x^{2}}{4} + \frac{x sin (2 x)}{4} + \frac{cos (2 x)}{8} + c$ $x^2/4+(xsin(2x))/4+cos(2x)/8+c$ ,where c is constant of integration.

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How do you integrate $x \cdot {cos}^{2} (x)$ $x * cos^2 (x)$ ?

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Explanation:

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