Integration by Subsitution

Example 01

\[\begin{aligned} \int_{1}^{2}xdx &= 4\int_{1}^{2}\frac{1}{2}x \cdot \frac{1}{2} dx \\ &= 4 \int_{1/2}^{1}t dt \end{aligned}\]

Example 02

\[\begin{aligned} \int_{-\infty}^{\infty}\frac{dy}{a^2+y^2} &= -j\int_{-\infty}^{\infty}\frac{jdy}{a^2-(jy)^2} \\ &= -j\int_{C}\frac{dz}{a^2-z^2} \end{aligned}\]

Example 03

\[X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt\] \[X(t) = \int_{-\infty}^{\infty} x(u)e^{-jut} du\] \[\omega = -u\]

san jian shi:

• bei ji han shu
• ji fen bian liang
• shang xia xian

\[X(t) = \int_{\dots}^{\dots} x(-\omega) e^{j\omega t} d(-\omega)\] \[X(t) = \int_{\dots}^{\dots} x(-\omega) e^{j\omega t} (-1) d\omega\] \[X(t) = \int_{\infty}^{-\infty} x(-\omega) e^{j\omega t} (-1) d\omega\]