Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec πt) sec πt tan πt dt

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Identify the integral to solve: \(\int e^{\sec(\pi t)} \sec(\pi t) \tan(\pi t) \, dt\).

Recognize that the integrand contains the function \(e^{\sec(\pi t)}\) multiplied by the derivative of \(\sec(\pi t)\), since the derivative of \(\sec(x)\) is \(\sec(x) \tan(x)\).

Use substitution by letting \(u = \sec(\pi t)\). Then, compute \(du\): since \(\frac{d}{dt} \sec(\pi t) = \pi \sec(\pi t) \tan(\pi t)\), it follows that \(du = \pi \sec(\pi t) \tan(\pi t) \, dt\).

Rewrite the integral in terms of \(u\) and \(du\): solve for \(dt\) in terms of \(du\) and substitute back into the integral to express it fully in terms of \(u\).

Integrate the resulting expression with respect to \(u\), then substitute back \(u = \sec(\pi t)\) to express the answer in terms of \(t\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, differentiating it, and rewriting the integral in terms of this variable. This technique is especially useful when the integral contains a function and its derivative.

Derivatives of Trigonometric Functions

Understanding the derivatives of trigonometric functions like secant and tangent is crucial. For example, the derivative of sec(x) is sec(x)tan(x), which often appears in integrals involving these functions. Recognizing these derivatives helps in identifying substitution candidates.

Exponential Functions with Composite Arguments

Exponential functions with composite arguments, such as e^(sec(πt)), require careful handling during integration. The chain rule applies when differentiating or integrating these functions, and recognizing the inner function and its derivative is key to simplifying the integral.

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Textbook Question

25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,

dy/dt = -0.6y

when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

Textbook Question

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Textbook Question

Evaluate the integrals in Exercises 53–76.

53. ∫dx/√(9-x²)