# Taking constants out of indefinite integrals

In the case of definite integrals, the linearity property implies that constants can be taken out of the integrals,

$$\int_{a}^{b} \alpha f(x) d x=\alpha \int_{a}^{b} f(x) d x \tag{1}$$

However, in the case of indefinite integrals, this leads to contradictory results in the case $$\alpha=0$$, since

$$\int 0\,dx = \int 0 \cdot 1 \,dx = 0 \int 1 \,dx = 0·(x+C) = 0$$

while the derivative of any constant equals $$0$$, so $$\int 0\,dx =C$$. Therefore, can't constants be taken out of indefinite integrals?

• I don't see a contradiction... $\int 0 = 0 + C$, no? Is that not what you have? Dec 11, 2021 at 21:01
• I ask the question in the sense that, when we take the zero out of the integral, the whole integral is multiplied by zero, including the constant of integration Dec 11, 2021 at 21:07
• What is the definition of $\int f$? What properties of that definition allow you to "factor out" constants? Read the definitions carefully, and I think you might spot your error. Dec 11, 2021 at 21:11

In $$\int \alpha f(x) dx$$ we are looking for the set of all functions such that the derivative of each one of them is $$\alpha f(x)$$. If $$F(x)$$ is such that $$F'(x)=f(x)$$ then for every $$\alpha$$ we have that $$(\alpha F)'(x)=\alpha f(x)$$ by the usual laws of derivatives. This is true even for $$\alpha =0$$.
However, if $$G(x)$$ is such that $$G'(x)=\alpha f(x)$$ then it does not in general follow that $$\left(\frac{G(x)}{\alpha}\right)'= f(x)$$. It does follow if $$\alpha\ne 0$$ (again by the usual laws of derivatives), but if $$\alpha=0$$ the left hand side is undefined. Therefore the set of all antiderivatives of $$\alpha f(x)$$ and the set $$\{\alpha F(x) | F'(x)=f(x)\}$$ is not the same set for $$\alpha=0$$. (More explicitly, the first set is all constants, the second set is just the zero function.)
• I am unaware of any calculus text that stipulates the condition $k\not=0$ when it writes the rule $\int kf(x)\,dx=k\int f(x)\,dx$. They probably should! Dec 13, 2021 at 15:45