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Chain Rule & Integration by Substitution

Review question

How could we integrate $\tan^4x$?
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Ref: R9297

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Question

Given that \(\dfrac{d}{dx}(\tan x)=\sec^2 x,\) differentiate \(\tan^3x\) with respect to \(x\).

Show that \[ \tan^4x=\tan^2x \sec^2x- \sec^2x+1 \] and hence evaluate \[ \int_0^{\pi/4}\tan^4x \, dx. \]