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

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

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integral of 2cos(x)cos(2x)

\int 2 cos (x) cos (2 x) d x

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

Solution steps

\int 2 cos (x) cos (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int cos (x) cos (2 x) d x

Apply Integration By Parts

: cos (2 x) sin (x) - \int - 2 sin (2 x) sin (x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

\frac{d}{d x} (arctan (0))

\frac{\partial}{\partial y} (\frac{x}{x + y})

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

x \to - 5 lim (\frac{x + 3}{x + 5})

\int 3 cos^{2} (t) d t

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