Answer
$$\int^4_{0}\frac{2t}{t^2-25}dt=\ln\frac{9}{25}$$
Work Step by Step
$$A=\int^4_{0}\frac{2t}{t^2-25}dt$$
We set $a=t^2-25$, which means $$da=2tdt$$
For $t=0$, $a=-25$ and for $t=4$, $a=-9$
Therefore, $$A=\int^{-9}_{-25} \frac{1}{a}da$$ $$A=\Big(\ln|a|\Big)\Big]^{-9}_{-25}$$ $$A=\ln9-\ln25=\ln\frac{9}{25}$$
