Differentiation and Integration

$$-n^2x$$  

$$n^2x$$  

$$-nx$$  

$$nx$$  

cos x

$$\cos x^2$$  

$$2x\ \cos x^2$$  

None of these

$$-4x\cos^2x^2\sin^2x^2$$  

$$-4x\cos^2x^2\sin x^2$$  

$$-2x\sin2x^2$$  

None of these

$$3\tan^2\left(4x^2\right)\sec^2\left(4x^2\right)$$  

$$12\tan^2\left(4x^2\right)\sec^2\left(4x^2\right)$$  

$$24\tan^2\left(4x^2\right)\sec^2\left(4x^2\right)$$  

$$24x\tan^2\left(4x^2\right)\sec^2\left(4x^2\right)$$  

$$\sin^{-1}x+C$$  

$$\tan x+C$$  

$$\tan^{-1}x+C$$  

$$\cot^{-1}x+C$$  

$$\frac{-x\ \sin2x}{2}+\frac{\cos2x}{4}+c$$  

$$\frac{-x\ \cos2x}{4}+\frac{\sin2x}{2}+c$$  

$$\frac{-x\ \cos2x}{2}+\frac{\sin2x}{4}+c$$  

$$-x\ \sin2x+\cos2x+c$$  

$$4$$  

$$5$$  

$$-3$$  

$$-2$$  

$$0,\ if\ f\left(-x\right)=f\left(x\right)$$  

$$2\int_0^af\left(x\right),\ if\ f\left(-x\right)=f\left(x\right)$$  

$$2\int_0^af\left(x\right),\ if\ f\left(-x\right)=-f\left(x\right)$$  

none of these

$$2\sqrt{\tan^2x+4}+c$$  

$$\log\ \sqrt{\tan^2x+4}+c$$  

$$\log\left|\tan\ x+\sqrt{\tan^2x+4}\right|+c$$  

none of these