What is the integral of (x-cos(x))^2 ?

The integral of (x-cos(x))^2 is (x^3)/3-2(xsin(x)+cos(x))+1/2 (x+1/2 sin(2x))+C

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integral of (x-cos(x))^2

\int (x - cos (x))^{2} d x

\frac{x ^{3}}{3} - 2 (x sin (x) + cos (x)) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + C

Solution steps

\int (x - cos (x))^{2} d x

Factor

x - cos (x) : - (- x + cos (x))

= \int (- (- x + cos (x)))^{2} d x

Simplify

(- (- x + cos (x)))^{2} : (- x + cos (x))^{2}

= \int (- x + cos (x))^{2} d x

Expand

(- x + cos (x))^{2} : x^{2} - 2 x cos (x) + cos^{2} (x)

= \int x^{2} - 2 x cos (x) + cos^{2} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x^{2} d x - \int 2 x cos (x) d x + \int cos^{2} (x) d x

\int x^{2} d x = \frac{x ^{3}}{3}

\int 2 x cos (x) d x = 2 (x sin (x) + cos (x))

\int cos^{2} (x) d x = \frac{1}{2} (x + \frac{1}{2} sin (2 x))

= \frac{x ^{3}}{3} - 2 (x sin (x) + cos (x)) + \frac{1}{2} (x + \frac{1}{2} sin (2 x))

Add a constant to the solution

= \frac{x ^{3}}{3} - 2 (x sin (x) + cos (x)) + \frac{1}{2} (x + \frac{1}{2} sin (2 x)) + C

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What is the integral of (x-cos(x))^2 ?
The integral of (x-cos(x))^2 is (x^3)/3-2(xsin(x)+cos(x))+1/2 (x+1/2 sin(2x))+C