Step 1:

The sum of Rs1000 is the principal or initial amount.($x_0)$

Step 2:

Let $x$ be the amount at time $t$ years,the amount being compounded continuously,at an annual rate of 4% per annum.

Step 3:

$\therefore$ rate of change of $x$ is 0.04

$\large\frac{dx}{dt}$$=0.04x$

$\large\frac{dx}{x}$$=0.04dt$

$\int \large\frac{dx}{x}$$=\int 0.04 dt+\log c$

$\log x=0.04t+\log c$

$\log\large\frac{x}{c}$$=0.04 t$

$\therefore x=ce^{0.04t}$

Step 4:

When $t=0,x_0=1000$

$x=1000e^{0.04t}$

Step 5:

When the amount doubles ,$x=2000$

$\therefore 2000=1000e^{0.04t}$

$\Rightarrow e^{0.04t}=2$

$0.04t\log_e e=\log _e 2$

$0.04t=0.6931$

$t=\large\frac{0.6931}{0.04}$

$t=17.3275$years

The time taken for the amount to double =17 years approximately.