Integrals

The value of the definite integral 02 13x+3 dx is

1. 12
2. 13
3. e 33
4. e 32

If = 122 -1x2x2-3x+2 dx, then the value of 7 ( 2 7 ) is . (Here, the inverse trigonometric function -1x assumes values in (- 2, 2 ).)

Let the function f:[1, ) R be defined by f(t)= cases(-1)n+12, & if t=2n-1, n N,\ (2n+1-t)2f(2n-1)+ (t-(2n-1))2f(2n+1), & if 2n-1<t<2n+1, n N. cases Define g(x)= 1xf(t) dt, x (1, ). Let denote the number of solutions of the equation g(x)=0 in the interval (1,8] and = x 1+ g(x)x-1. Then the value of + is equal to .

The value of 2 0 2f(x)g(x) dx- 0 2g(x) dx is .

The value of 16 3 0 2f(x)g(x) dx is .