Dayaan M. answered • 08/26/26
Earned A’s in Calc 1/AB & Calc 2/BC | 5 Years of Tutoring Experience
This one is false, and it is worth understanding why, because it is a mistake that is very easy to make.
Integration does play nicely with addition and subtraction. The integral of f(x) + g(x) really is the integral of f plus the integral of g, and the same works for subtraction. Because those rules exist, students often assume the same thing must be true for multiplication and division. It is not.
The fastest way to convince yourself is to try a small example where you can compute both sides. Let f(x) = x and g(x) = x, and integrate from 1 to 2.
The left side is the integral of f(x)/g(x), which is the integral of x/x. That simplifies to the integral of 1 from 1 to 2, and the area under the constant 1 across a width of 1 is:
left side = 1
Now the right side. The integral of x from 1 to 2 is x^2/2 evaluated from 1 to 2, which is 2 - 1/2 = 3/2. The bottom integral is exactly the same thing, so it is also 3/2. Dividing:
right side = (3/2)/(3/2) = 1
Well, those happened to match, so let us pick a case that separates them. Take f(x) = x and g(x) = 1, again from 1 to 2. The left side is the integral of x/1 = x, which we just found is 3/2. The right side is the integral of x divided by the integral of 1, which is (3/2)/1 = 3/2. Still equal, only because g was so simple.
Now try f(x) = x and g(x) = x^2 from 1 to 2. The left side is the integral of x/x^2 = 1/x, which is ln(2) - ln(1) = ln(2), about 0.693. The right side is the integral of x over the integral of x^2, which is (3/2) divided by (8/3 - 1/3) = (3/2)/(7/3) = 9/14, about 0.643. Those are not equal.
So, our final answer is FALSE. There is no quotient rule for integrals, and there is no product rule either. When you have a fraction inside an integral, you have to actually simplify it or use a real technique like u-substitution or partial fractions.
