3.9: Trigonometric derivative identity and integration

0606/22/F/M/24 — Question 4 · 12 marks

(a)(i)

Given that

y = 3\sin^{2} x + \cos x

, show that

\cot x\,\dfrac{\mathrm{d}y}{\mathrm{d}x} + y = k(1 + \cos^{2} x)

, where

k

is an integer.

[4]

(a)(ii)

Using your value of

k

, solve

k(1 + \cos^{2} x) = 4

for

-\pi \leqslant x \leqslant \pi

.

[4]

(b)(i)

Differentiate

y = x\tan x

with respect to

x

.

[2]

(b)(ii)

Hence find

\displaystyle\int\left(\tan x + \dfrac{x}{\cos^{2} x}\right)\,\mathrm{d}x

.

[2]