Show that $\dfrac{d}{dx}(\ln(\cos x))=-\tan x$.[2] Differentiate $x\tan x$ with respect to

Question

Show that $\dfrac{d}{dx}(\ln(\cos x))=-\tan x$.[2] Differentiate $x\tan x$ with respect to $x$.[2] Using the results from part (a) and (b), find $\int x\sec^2 x\,dx$ and hence show that $\int_0^{\frac{\pi}{4}} x\sec^2 x\,dx=\dfrac{\pi}{4}-\dfrac{1}{2}\ln 2$.[4] Use the logarithm identity and indefinite antiderivative on intervals where $\cos x>0$. This includes the whole definite-integration interval $[0,\pi/4]$.

Accepted Answer

$(a)\ -\tan x$ $(b)\ x\sec^2x+\tan x$ $(c)\ \int x\sec^2x\,dx=x\tan x+\ln(\cos x)+C,\ \int_0^{\frac{\pi}{4}}x\sec^2x\,dx=\dfrac{\pi}{4}-\dfrac{1}{2}\ln2$

2025 AISS PRELIMS P1 Q4

2025 AISS PRELIMS P1 Q4

Solution:

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