# 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 #