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### Course: AP®︎/College Calculus BC>Unit 6

Lesson 8: Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule

# Indefinite integrals: sums & multiples

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.
• can someone explain to me how the reverse power rule works for constant numbers like 8 or 9?
(1 vote)
• Imagine 8 as 8x^0. Integrating this using the reverse power rule, we get [8x^(0+1)]/(0+1) = 8x. You can verify that 8x is the antiderivative of 8 by differentiating 8x.
• at , if it were definite integral, then it makes sense. so i assume this property applies to indefinite integral too
• Indeed! Most properties of integrals are common between a definite and indefinite integral. It's just that in the former, you have to do an extra step of plugging in numbers, while that doesn't need to be done in the latter.
• 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