Evaluate the integrals in Exercises 31–78.
41. ∫(from 0 to 4)2t/(t² - 25)dt

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Identify the integral to be evaluated: $\int_0^4 \frac{2t}{t^2 - 25} \, dt$.

Notice that the numerator \$2t$ is the derivative of the denominator $t^2 - 25$, which suggests using the substitution method.

Let $u = t^2 - 25$, then compute $du = 2t \, dt$. This means $2t \, dt = du$.

Change the limits of integration according to the substitution: when $t=0$, $u = 0^2 - 25 = -25$; when $t=4$, $u = 4^2 - 25 = 16 - 25 = -9$.

Rewrite the integral in terms of $u$: $\int_{-25}^{-9} \frac{1}{u} \, du$, which can be integrated using the natural logarithm function.

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