1-
Solve for x over the real numbers:
3 x^2-1/2 = 0
Add 1/2 to both sides:
3 x^2 = 1/2
Divide both sides by 3:
x^2 = 1/6
Take the square root of both sides:
Answer: | x = 1/sqrt(6) or x = -1/sqrt(6)
2-
Take the integral:
integral (4 x^3-3 x+2) dx
Integrate the sum term by term and factor out constants:
= 4 integral x^3 dx-3 integral x dx+2 integral 1 dx
The integral of x^3 is x^4/4:
= x^4-3 integral x dx+2 integral 1 dx
The integral of x is x^2/2:
= -(3 x^2)/2+x^4+2 integral 1 dx
The integral of 1 is x:
Answer: | = x^4-(3 x^2)/2+2 x+constant
3-
Take the integral:
integral (x y+2 x) dx
Integrate the sum term by term and factor out constants:
= (y+2) integral x dx
The integral of x is x^2/2:
Answer: | = 1/2 x^2 (y+2)+constant
4-
Take the integral:
integral (-4/x^3+1/x^2+2) dx
Integrate the sum term by term and factor out constants:
= integral 1/x^2 dx-4 integral 1/x^3 dx+2 integral 1 dx
The integral of 1/x^2 is -1/x:
= -1/x-4 integral 1/x^3 dx+2 integral 1 dx
The integral of 1/x^3 is -1/(2 x^2):
= 2/x^2-1/x+2 integral 1 dx
The integral of 1 is x:
= 2/x^2+2 x-1/x+constant
Which is equal to:
Answer: | = (2 x^3-x+2)/x^2+constant
5-
Take the integral:
integral (-x^2-2/x^2-x+1) dx
Integrate the sum term by term and factor out constants:
= - integral x^2 dx- integral x dx-2 integral 1/x^2 dx+ integral 1 dx
The integral of x^2 is x^3/3:
= -x^3/3- integral x dx-2 integral 1/x^2 dx+ integral 1 dx
The integral of x is x^2/2:
= -x^2/2-x^3/3-2 integral 1/x^2 dx+ integral 1 dx
The integral of 1/x^2 is -1/x:
= 2/x-x^2/2-x^3/3+ integral 1 dx
The integral of 1 is x:
Answer: | = -x^3/3-x^2/2+x+2/x+constant
