The value of _0^2[2x(1+3x)],dx, where [t] denotes the greatest integer function, is:

Question

The value of $\displaystyle\int_0^{2\pi}[\sin2x(1+\cos3x)]\,dx$, where $[t]$ denotes the greatest integer function, is:

Accepted Answer

$$I = \int\limits_0^{2\pi } {\left[ {\sin 2x(1 + \cos 3x)} \right]} dx$$ .... (i)

$$ \therefore\int\limits_0^a {f(x)}  = \int\limits_0^a {f(a - x)} dx$$

$$ \because I = \int\limits_0^{2\pi } {\left[ { - \sin 2x(1 + \cos 3x)} \right]} dx$$

By (i) + (ii)

$$ \Rightarrow 2I = \int\limits_0^{2\pi } {( - 1)dx} $$

$$ \Rightarrow 2I =  - (x)_0^{2\pi }$$

$$ \Rightarrow $$ I = --$$\pi $$

Solutions · 1