If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Integral Calculus

### Course: Integral Calculus>Unit 1

Lesson 10: Reverse power rule

# Indefinite integrals: sums & multiples

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.C (LO)
,
FUN‑6.C.1 (EK)
,
FUN‑6.C.2 (EK)
An indefinite integral of a sum is the same as the sum of the integrals of the component parts. Constants can be "taken out" of integrals.

## Want to join the conversation?

• Sal explained that the definite integral is the area under the curve from a to b (a is an under bound and b is an upper bound). However, there is no such thing as under bound or upper bound in the indefinite integral. Then what does indefinite integral refer to in the graph?
• Suppose we have a function, f(x). The indefinite integral of the function will be another function, F(x), such that F(c) is equal to the area under the curve generated by f(x) between x=0 and x=c.
• When we are dealing with integrals and using the symbol ∫, how am I going to know if it is a definite integral or an indefnite integral? Is there some way to distinguish the two?
(1 vote)
• Definite integrals have bounds of integration written on the top and bottom of the integral symbol while indefinite integrals do not.
• If we have a function like
`4x+7` and then we take the anti derivative, shouldn't this mean the 4x becomes `2x^2 + c` and the 7 becomes `7x+c` causing the final equation to be `2x^2 + 7x +2c`?
• c is an arbitrary constant, so multiplying it by another constant does not matter and we can remove the factor of 2.
• Definite integrals have the same properties as indefinite integrals, don't they ?
• If you're algebraically doing integration, these properties will work with either type.
• Is this really a proof of the two properties? I can't see how.
• in is it the the fundumental theorom that sal is applying?because there is a constant there and why are we allowed to treat constant like that?
• That is not the fundamental theorem of calculus. Sal showed us that property in this video, so you probably didn't know it beforehand.
(1 vote)
• If I'm asked to find the antiderivative of 4x+7, is that sum rule? I'm not sure because its a variable plus a constant rather than adding two variables. But I don't know what else to do...
(1 vote)
• This late reply is for others who may have the same question. It's a very good question. I am just a student here myself, but perhaps I can help. To me, it is useful to think of 7 as 7x^0.
x^0 = 1 and 1 * 7 = 7; therefore 7 = 7x^0.
You could then apply the sum rule and the reverse power rule.
int.(4x + 7) dx = (4x^(1+1))/(1+1) + C1 + (7x^(0+1))/(0+1) + C2
= (4x^2)/2+ C1 + (7x^1)/1 + C2 = 2x^2 + 7x + C
• What's the difference between definite integrals and indefinite integrals ?
(1 vote)
• An indefinite integral results in a set of functions whose derivatives are equal to the integrand.
∫𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶
𝐹 '(𝑥) = 𝑓(𝑥)

A definite integral is when we evaluate 𝐹(𝑏) − 𝐹(𝑎), which gives us the area under 𝑓(𝑥) over the interval [𝑎, 𝑏].
∫[𝑎, 𝑏] 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)