# proving that a certain function defined using integrals is continuous

How can I prove that the function

$$f(x)=\int_{0}^{\pi}\frac{\sin(xt)}{t}dt$$

I'm very confused here, because the variable "x" appears inside the integral, and not in the form $\int_{0}^{x}$ as always, I don't know how can I do here. Please help me!

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Treat $x$ as constant. Because we are integrating with respect to $t$. – Inceptio May 29 '13 at 7:22

## 1 Answer

Hint: $$\int\limits_0^\pi\frac{\sin(xt)}{t}dt =\int\limits_0^\pi\frac{\sin(xt)}{xt}d(xt) =\int\limits_0^{\pi x}\frac{\sin(s)}{s}ds$$

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I did not understand what you did – Trafalgar Law May 30 '13 at 3:06
Another attempt, try change of variables $s=xt$ – Norbert May 30 '13 at 4:37