The function f whose graph passes through $$(0$$ ,$$0$$ $$)and$$ whose derivative is $$cos4$$⁡$$x+sin4$$⁡$$xcos2$$⁡x−$$sin2$$⁡x is given by

Question

The function f whose graph passes through $$(0$$ ,$$0$$ $$)and$$ whose derivative is $$cos4$$⁡$$x+sin4$$⁡$$xcos2$$⁡x−$$sin2$$⁡x is given by

Infinity Learn · Accepted Answer

We have $$f(x)=$$∫$$cos4$$⁡$$x+sin4$$⁡$$xcos2$$⁡x−$$sin2$$⁡xdxUsing, $$2cos4$$⁡$$x+sin4$$⁡$$x=cos2$$⁡$$x+sin2$$⁡$$x2+cos2$$⁡x−$$sin2$$⁡$$x2$$, we can write  $$f(x)=12$$∫$$1cos2$$⁡x−$$sin2$$⁡xdx                               $$+12$$∫$$cos2$$⁡x−$$sin2$$⁡$$xdx=12$$∫$$sec$$⁡$$2xdx+12$$∫$$cos$$⁡$$2xdx=14log$$⁡$$tan$$⁡$$x+$$π$$4+14sin$$⁡$$2x+C=14log$$⁡$$1+$$$$tan$$⁡$$x1$$−$$tan$$⁡$$x+12sin$$⁡xcos⁡$$x+CSince$$ $$f(0)=0$$ , we get $$C=$$ $$0$$.

The function $f$ whose graph passes through (0 ,0 )
and whose derivative is $\frac{\cos^{4} x + \sin^{4} x}{\cos^{2} x - \sin^{2} x}$ is given by

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The function f whose graph passes through (0 ,0 )and whose derivative is cos4⁡x+sin4⁡xcos2⁡x−sin2⁡x is given by

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The function $f$ whose graph passes through (0 ,0 )
and whose derivative is $\frac{\cos^{4} x + \sin^{4} x}{\cos^{2} x - \sin^{2} x}$ is given by