Using sinc function we write the well-known Dirichlet's integral. We define the norm on $L^1$ spaces and prove the monotone convergence theorem, Fatou's lemma, Lebesgue's dominated convergence theorem and that functions $C(\mathbb{R}) \cap L^1(\mathbb{R})$ form a dense subspace of $L^1(\mathbb{R})$. We prove Riemann-Lebesgue's lemma and Fourier inversion theorem. We define convolution and prove that it is associative and commutative and using convolution and Fourier transform we derive the formula of an integral with borders $-\infty$ and $\infty$ of a product of a finite amount of functions. We calculate the Fourier transform of the function sinc and using that we calculate the values of some of the Borwein integrals and find an upper bound for others. We then calculate the value of the Borwein integral that we couldn't calculate before with the previous methods. Using the Lebesgue's dominated convergence theorem we study the behaviour of the rest of the values of Borwein integrals.
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