Write an anti derivative for each of the following functions using the method of inspection:

(i) $\cos 2x$

(ii) $3x^{2} + 4x^{3}$

(iii) $\frac{1}{x}, x \neq 0$

(i) We look for a function whose derivative is $\cos 2x$. Recall that

$$\frac{d}{dx} \sin 2x = 2 \cos 2x$$

or $\cos 2x = \frac{1}{2} \frac{d}{dx} (\sin 2x) = \frac{d}{dx} \left(\frac{1}{2} \sin 2x\right)$

Therefore, an anti derivative of $\cos 2x$ is $\frac{1}{2} \sin 2x$.

(ii) We look for a function whose derivative is $3x^{2} + 4x^{3}$. Note that

$$\frac{d}{dx} \left(x^{3} + x^{4}\right) = 3x^{2} + 4x^{3}.$$

Therefore, an anti derivative of $3x^{2} + 4x^{3}$ is $x^{3} + x^{4}$.

(iii) We know that

$$\frac {d}{d x} (\log x) = \frac {1}{x}, x > 0 \text{ and } \frac {d}{d x} [ \log (- x) ] = \frac {1}{- x} (- 1) = \frac {1}{x}, x < 0$$

Combining above, we get $\frac{d}{dx} (\log |x|) = \frac{1}{x}, x \neq 0$

Therefore, $\int \frac{1}{x} dx = \log |x|$ is one of the anti derivatives of $\frac{1}{x}$ .