The integral

The value of the integral $$\int_0^{\infty} e^{-x^2} \, dx$$ is​

What is wrong with the following calculation?

​$$\int_{-1}^3 \frac{1}{x^2} d x=-\frac{4}{3}$$​​

​$$\int_{0}^{\frac{\pi}{4}} \sin^3 \theta \, d\theta = \, ?$$​​

Let  $$\begin{aligned}3f(x) + f\left(\frac{1}{x}\right) = \frac{1}{x} + 1\end{aligned}$$ Then what is $$8 \int_{1}^{2} f(x) \, dx$$ equal to?