Subtask 1

´int 1/(x^2-1) dx´

Factor ´-1´ from the denominator: ´= - int 1/(1-x^2) dx´

The integral of ´1/(1-x^2)´ is ´tanh^(-1)(x)´: ´= -tanh^(-1)(x) + "constant"´

For ´x in RR ^^ -1 < x < 1´this is equivalent to: ´= 1/2 (log(1-x)-log(x+1)) + "constant"´

Subtask 2

´int sin(8 x+4) dx´

For the integrand ´sin(8 x+4)´, substitute ´u = 8 x + 4´ and ´du = 8 dx´: ´= 1/8 int sin(u) du´

The integral of ´sin(u)´ is ´-cos(u)´: ´= -(cos(u))/8 + "constant"´

Substitute back for ´u = 8 x+4´: ´= -1/8 cos(8 x+4) + "constant"´

Subtask 3

´int e^x x^2 dx´

For the integrand ´e^x x^2´, integrate by parts, ´int f dg = f g- int g df´, where ´f = x^2´, ´dg = e^x dx´, ´df = 2 x dx´, ´g = e^x´: ´= e^x x^2-2 int e^x x dx´

For the integrand ´e^x x´, integrate by parts, integral ´f dg = f g- int g df´, where ´f = x´, ´dg = e^x dx´, ´df = dx´, ´g = e^x´: ´= e^x x^2-2 e^x x+2 int e^x dx´

The integral of e^x is e^x: ´= e^x x^2-2 e^x x+2 e^x + "constant"´

´= e^x (x^2-2 x+2) + "constant"´