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I'm studying integrals and primitives and I have this enormous doubt about whether or not write and arbitrary constant $C$... And should I determine that constant.

See an example:

Discover the only diferenttiable function $f: \mathbb{R} \longrightarrow \mathbb{R}$ (not $=0$) that satisfies the equation

$$\int_0^x\left((f(x)^3\right)\,dx=\left(\int_0^x(f(x)\,dx\right)^2$$

So we determine $f$ and we reach to $$f(x) = x + C$$ and the the solution simply says $C=0$? How do we know that? How do we know when to add or not the constant? I know that if we know the value of the primitive in a certain point but that0s not the case, right?

Con someone please clarify this "arbitrary constant" thing to me? Thanks!

AugSB
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Granger Obliviate
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1 Answers1

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The arbitrary constant has to be added whenever you are dealing with indefinite integrals. In other words, if you are told that $F$ is the antiderivative of a function $f$, meaning that $$F(x) = \int f(x) dx \qquad\mbox{ (integral without limits)},$$ then any other funtion $G(x)=F(x)+C$ will be an antiderivative of $f$ too. In fact, $$F'(x) = G'(x) = f(x).$$

Now, in your case, $f$ is not an antiderivative. You are asked you find a function that satisfies that identity. Note that if you substitute $f(x)=x+C$ in it, the only way that the equality becomes true is that $C=0$:

$$\left.\begin{array}{r} \int_0^x (x+C)^3\,dx = \frac{(x+C)^4}{4} - \frac{C^4}{4} \\ \left(\int_0^x (x+C)\,dx\right)^2 = \left(\frac{x^2}{2}+C x\right)^2 \end{array}\right\}\longrightarrow \frac{(x+C)^4}{4} - \frac{C^4}{4} = \left(\frac{x^2}{2}+C x\right)^2 \Leftrightarrow C=0. $$

AugSB
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