I'm sorry to say I don't understand any of the following question. Hoping someone can break it down with a solution. (and if anyone can suggest the best free online integral calculator with solutions, that would be great. Wouldn't mind paying for it but all the paid ones have terrible reviews)
Let $\left(f_{n}\right)_{n \in \mathbb{N}}$ be a squence of functions with $f_{n}:[0, \pi] \rightarrow \mathbb{R}$ defined as
$$ f_{n}(x)=\left\{\begin{array}{ll} n \sin (n x) & \text { for } 0 \leq x \leq \frac{\pi}{n} \\ 0 & \text { for } \frac{\pi}{n} \leq x \leq \pi \end{array}, \quad n \in \mathbb{N}\right. $$
a) Show that $f_{n}$ is integrable for all $n \in \mathbb{N}$.
b) Show that the sequence $f_{n}(x)$ converges pointwise for every $x \in[0, \pi]$.
c) Does $f_{n}$ fulfill the requirements to apply the Lemma of Fatou and the Dominated Convergence Theorem, respectively?
d) Does the equality of
$$ \lim _{n \rightarrow \infty} \int_{[0, \pi]} f_{n} d \mu=\int_{[0, \pi]} \lim _{n \rightarrow \infty} f_{n} d \mu $$
hold?