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

The integral of (cos(x)+2cos(2x))^2 is 5/2 x+1/4 sin(2x)+1/2 sin(4x)+16/3 sin^3(x)+4cos(2x)sin(x)+C

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

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

\frac{5}{2} x + \frac{1}{4} sin (2 x) + \frac{1}{2} sin (4 x) + \frac{16}{3} sin^{3} (x) + 4 cos (2 x) sin (x) + C

Solution steps

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

Expand

(cos (x) + 2 cos (2 x))^{2} : cos^{2} (x) + 4 cos (x) cos (2 x) + 4 cos^{2} (2 x)

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

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

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

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

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

Simplify

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 4 (cos (2 x) sin (x) + \frac{4}{3} sin^{3} (x)) + 2 (x + \frac{1}{4} sin (4 x)) : \frac{5}{2} x + \frac{1}{4} sin (2 x) + \frac{1}{2} sin (4 x) + \frac{16}{3} sin^{3} (x) + 4 cos (2 x) sin (x)

= \frac{5}{2} x + \frac{1}{4} sin (2 x) + \frac{1}{2} sin (4 x) + \frac{16}{3} sin^{3} (x) + 4 cos (2 x) sin (x)

Add a constant to the solution

= \frac{5}{2} x + \frac{1}{4} sin (2 x) + \frac{1}{2} sin (4 x) + \frac{16}{3} sin^{3} (x) + 4 cos (2 x) sin (x) + C

f (x) = 10 x sin (x), at x = \frac{π}{2}

\frac{\partial}{\partial x} (\frac{1}{4 - x ^{2}})

x \to 0 lim (\frac{x ^{2}}{x ^{4}})

\frac{d}{d x} (x^{3} - 3 x^{2} - 8 x + 2)

\frac{d}{d x} (\frac{3 x ^{2}}{2 x})

What is the integral of (cos(x)+2cos(2x))^2 ?
The integral of (cos(x)+2cos(2x))^2 is 5/2 x+1/4 sin(2x)+1/2 sin(4x)+16/3 sin^3(x)+4cos(2x)sin(x)+C