3.6.1 Integration (A-level only)

a.

Show that the trigonometric identity

\frac{\sin 2x}{1 + \tan^2 x} \equiv 2 \sin x \cos^3 x

is valid for all $x\,$ where the expression is defined.

[3]

b.

In a wave mechanics simulation, the rate of change of energy density $E\,$ with respect to phase $ϕ\phi$ (in radians) is modeled by the equation:

\frac{dE}{d\phi} = \frac{30 \sin 6\phi}{1 + \tan^2 3\phi}

Hence, determine the general expression for $E(ϕ)E(\phi)$ .

[6]

a.

Show that the trigonometric identity

\frac{\sin 2x}{1 + \tan^2 x} \equiv 2 \sin x \cos^3 x

is valid for all $x\,$ where the expression is defined.

[3]

b.

In a wave mechanics simulation, the rate of change of energy density $E\,$ with respect to phase $ϕ\phi$ (in radians) is modeled by the equation:

\frac{dE}{d\phi} = \frac{30 \sin 6\phi}{1 + \tan^2 3\phi}

Hence, determine the general expression for $E(ϕ)E(\phi)$ .

[6]

3.6.1 Integration (A-level only) Questions