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

The integral of (x+cos(2x))^2 is (x^3)/3+xsin(2x)+1/2 cos(2x)+1/2 (x+1/4 sin(4x))+C

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

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

\frac{x ^{3}}{3} + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{1}{2} (x + \frac{1}{4} sin (4 x)) + C

Solution steps

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

Expand

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

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

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

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

\int cos^{2} (2 x) d x = \frac{1}{2} (x + \frac{1}{4} sin (4 x))

= \frac{x ^{3}}{3} + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{1}{2} (x + \frac{1}{4} sin (4 x))

Add a constant to the solution

= \frac{x ^{3}}{3} + x sin (2 x) + \frac{1}{2} cos (2 x) + \frac{1}{2} (x + \frac{1}{4} sin (4 x)) + C

\int \frac{2 x ^{3}}{x + 1} d x

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What is the integral of (x+cos(2x))^2 ?
The integral of (x+cos(2x))^2 is (x^3)/3+xsin(2x)+1/2 cos(2x)+1/2 (x+1/4 sin(4x))+C