Exercises

Follow the instructions for the exercises below

Use Commands Only

1. When x is less than -2, substitute y=$\frac{x}{x^2-2x+4}$ into $\frac{1}{2}\frac{1+2y+\sqrt{1+4y-12y^2}}{y}$
   (x should be the result)

   restart
assume(x<-2)
eq1:=1/2*(1+2*y+sqrt(1+4*y-12*y^2))/y
y:=x/(x^2-2*x+4)
eq1
simplify(eq1)

2. Make the appropriate assumption so that the result of ${\displaystyle \underset{0}{\overset{\pi }{\int }}\cos (nx)dx}$ is 0.

   restart
assume(n, integer)
eq2:=cos(n*x)
int(eq2, x=0..Pi)

3. When n is real, take the integral with respect to x and simplify:
   $\frac{{\left(1+x^{\left(\frac{1}{4}\right)}\right)}^n}{\sqrt{x}}$

   restart
assume(n, real)
eq3:=((1+x^(1/4))^n)/sqrt(x)
inteq3:=int(eq3, x)
simplify(inteq3)

4. If p is greater than 0, integrate with respect to x from 1 to $\infty$
   $(a{x}^2+bx+c)e^{-px}$

   restart
assume(p>0)
eq4:=(a*x^2+b*x+c)*exp(-p*x)
inteq4:= int(eq4, x=1..infinity)
simplify(inteq4)

Use the Right Click Menu

1. When n is real, take the integral with respect to x and simplify:
   $\frac{{\left(1+x^{\left(\frac{1}{4}\right)}\right)}^n}{\sqrt{x}}$

   restart
assume(n, real)
eq3:=((1+x^(1/4))^n)/sqrt(x)
(Right-click: Integrate->x)
(Right-click result: simplify)