Show that int_0^1sinx/sqrt(x^2+1)dx<sqrt2-1 ?
1 Answer
See explanation
Explanation:
We want to show
int_0^1sin(x)/sqrt(x^2+1)dx < sqrt(2)-1
This is a quite "ugly" integral, so our approach will not be to solve this integral, but compare it to a "nicer" integral
We now that for all positive real numbers
Thus, the value of the integrand will also be bigger, for all positive real numbers, if we substitute
int_0^1x/sqrt(x^2+1)dx < sqrt(2)-1
Then our first statement must also be true
The new integral is a simple substitution problem
int_0^1x/sqrt(x^2+1)=[sqrt(x^2+1)]_0^1=sqrt(2)-1
The last step is to notice that
Therefore we can conclude
int_0^1sin(x)/sqrt(x^2+1)dx < sqrt(2)-1