\pi\end{array}\right." alt="\int\limits_{-\infty}^{\infty}f(x)dx=1\\\int\limits_{0}^\pi a\sin xdx=a\int\limits_{0}^\pi \sin xdx=a\left.(-\cos x)\right|\limits_{0}^\pi=a(-0+1)=a=1\\a=1\\f(x)=\left\{\begin{array}{lll}0&npu&x<0\\\sin x&npu&0\leq x\leq\pi\\0&npu&x>\pi\end{array}\right." align="absmiddle" class="latex-formula">
\pi\Rightarrow f(x)=0\Rightarrow F(x)=\int\limits_{-\infty}^00dt+\int\limits_0^\pi\sin tdt+\int\limits_\pi^x0dt=\left.\left(-\cos t\right)\right|_0^\pi=\\=-0+1=1" alt="F(x)=\int\limits_{-\infty}^xf(t)dt\\x<0\Rightarrow f(x)=0\Rightarrow F(x)=\int\limits_{-\infty}^x0dt=0\\0\leq x\leq\pi\Rightarrow f(x)=\sin x\Rightarrow F(x)=\int\limits_{-\infty}^00dt+\int\limits_0^x\sin tdt=\\=0+\left.(-\cos t)\right|\limits_0^x=-\cos x+\cos0=1-\cos x\\x>\pi\Rightarrow f(x)=0\Rightarrow F(x)=\int\limits_{-\infty}^00dt+\int\limits_0^\pi\sin tdt+\int\limits_\pi^x0dt=\left.\left(-\cos t\right)\right|_0^\pi=\\=-0+1=1" align="absmiddle" class="latex-formula">
