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I have another integral that I have to evaluate in the context of measure and integration theory. Again, this might be done with MCT or DCT. I still got my problems with trigonometric factors though.

The task is to evaluate $$\lim_{n\to \infty} \int_{[0, \pi/4]} \text{sin}(nx)\,\text{log}(1+x^n)\, d\lambda(x)$$

i.) Pointwise convergence: Let $$f_n(x) := \text{sin}(nx)\,\text{log}(1+x^n)$$ Since $x \in [0, \pi/4]$, it holds that $x < 1$ and hence $\lim_{n\to\infty} x^n = 0$. Because of the continuity of the logarithm, it holds that $$\lim_{n\to\infty} \text{log}(1 + x^n) = \text{log}(\lim_{n\to\infty} 1 + x^n) = \text{log}(1) = 0$$ But how should I handle the trigonometric term? For $\lim_{n\to\infty} \text{sin}(nx)$ jumps around.

ii.) Dominating function: I have to find a non-negative function $g \in L^1(\mu)$ s.t. $$\vert f_n(x)\vert \leq g(x)$$ for all $x \in [0, \pi/4]$. Can I just take $g(x) = x$ here?

iii.) Application of DCT: Due to the validity of the prerequisites I can now apply the DCT and set $$\lim_{n\to \infty} \int_{[0, \pi/4]} \text{sin}(nx)\,\text{log}(1+x^n)\, d\lambda(x) = \int_{[0, \pi/4]} \lim_{n\to \infty} \text{sin}(nx)\,\text{log}(1+x^n)\, d\lambda(x)$$. The problem now is that I still need the limit from i.) in order to complete the integration.

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    Use the Riemann-Lebesgue lemma after bounding the integrands uniformly by $\log(1+x)$. – 2017-02-14

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The problem now is that I still need the limit from i.) in order to complete the integration.

Hint. To prove your $i)$ statement, use $$ \left|f_n(x)\right|= \left|\text{sin}(nx)\,\text{log}(1+x^n)\right|\le\left|\text{log}(1+x^n)\right|\le |x|^n,\quad |x|\le\frac \pi4<1, $$ then apply the DCT as wanted.

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    But how does that prove my statement? Doesn't it just prove that $\vert x\vert^n$ is a dominating function? – 2017-02-14
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    @Taufi It does prove your statement since it gives, for $|x|\le\pi/4<1$, $$\lim_{n \to \infty} \left|\text{sin}(nx)\,\text{log}(1+x^n)\right|\le\lim_{n \to \infty} \left|x\right|^n=0.$$ The dominating function can be $1$. – 2017-02-14
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    Oh, you are right. I missed that! I accept your answer as it is closer to the DCT procedure I have to follow. – 2017-02-14
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I suggest that you do not use any fancy theorems (if you must, then follow Olivier's tip), but rather just estimate, using the triangle inequality and $\log(1+t)\leq t$, $$ \Bigl|\int_0^{\pi/4}\sin(n x)\log(1+x^n)\,dx\Bigr|\leq \int_0^{\pi/4}\log(1+x^n)\,dx\leq \int_0^{\pi/4}x^n\,dx=\frac{1}{n+1}\Bigl(\frac{\pi}{4}\Bigr)^{n+1}\to 0. $$

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    Thanks. Tell that my professor, but you are right: once I know some tools I want to apply them everywhere, even if not necessary. – 2017-02-14