Question

Write an anti-derivative for the functions using the method of inspection:
$\left(i\right)\cos 2x$
$\left(ii\right)3x^2+4x^3$
$\left(iii\right)\frac{1}{x},x\ne 0$

Answer 1

$\left(i\right)$ We are looking for a function whose derivative is $\cos 2x$
We can recall that,
$\frac{d}{dx}\sin 2x=2\cos 2x$
$\Rightarrow \cos 2x=\frac{1}{2}\frac{d}{dx}\left(\sin 2x\right)$
$\Rightarrow \cos 2x=\frac{d}{dx}\left(\frac{1}{2}\sin 2x\right)$
Hence, the anti-derivative of $\cos 2x$ is $\frac{1}{2}\sin 2x$

$\left(ii\right)$ We are looking for a function whose derivative is $3x^2+4x^3$
As we know,
$\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$
$\therefore \frac{d}{dx}(x^3+x^4)=3x^2+4x^3$
Hence, the anti-derivative of $3x^2+4x^3$ is $x^3+x^4$

$\left(iii\right)$ We know that
$\Rightarrow \frac{d}{dx}\left(\log x\right)=\frac{1}{x},\text{when}x>0\cdots \left(i\right)$
and
$\Rightarrow \frac{d}{dx}\left[\log \right(-x\left)\right]=\frac{1}{-x}(-1)=\frac{1}{x},\text{when}x<0\cdots \left(ii\right)$
Combining above equations, we get
$\Rightarrow \frac{d}{dx}\left(\log |x|\right)=\frac{1}{x},x\ne 0$
Hence, the anti-derivatives of $\frac{1}{x}$ is $\left(\log |x|\right)$