How do you integrate {x(sqrt(1+x^2)} dx{x(1+x2}dx from 0 to 1?

1 Answer
Sep 22, 2016

I=1/3(2sqrt2-1)I=13(221).

Explanation:

Let I=int_0^1 {xsqrt(1+x^2)}dx.I=10{x1+x2}dx.

We subst. 1+x^2=t^2 rArr 2xdx=2tdt, i.e., xdx=tdt1+x2=t22xdx=2tdt,i.e.,xdx=tdt

Further, x=0 rArr t=1, and, x=1 rArr t=sqrt2x=0t=1,and,x=1t=2.

:. I=int_1^(sqrt2) t.tdt = int_1^(sqrt2) t^2dt=[t^3/3]_1^sqrt2.

=1/3[sqrt2^3-1].

:. I=1/3(2sqrt2-1).