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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Definite Integrals

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫ from a to b of f(t) dt, where a and b are the lower and upper bounds. Evaluating definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Integration Techniques - Substitution

Substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integrand contains a function and its derivative.

Handling Rational Functions in Integration

Rational functions are ratios of polynomials, and integrating them often requires algebraic manipulation like partial fraction decomposition or substitution. Recognizing the structure of the numerator and denominator helps in choosing the right method to simplify and evaluate the integral.

Evaluate the integrals in Exercises 31–78.41. ∫(from 0 to 4)2t/(t² - 25)dt