Evaluate the integrals in Exercises 37–46.

∫ √t sin(2t³/²)dt

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Identify the integral to solve: \(\int \sqrt{t} \sin\left(2t^{3/2}\right) \, dt\).

Rewrite the integrand to recognize substitution possibilities. Note that \(\sqrt{t} = t^{1/2}\) and the argument of sine is \(2t^{3/2}\), which suggests a substitution involving \(t^{3/2}\).

Let \(u = 2t^{3/2}\). Then compute \(du\): differentiate \(u\) with respect to \(t\) using the chain rule: \(\frac{du}{dt} = 2 \cdot \frac{3}{2} t^{1/2} = 3 t^{1/2}\). Therefore, \(du = 3 t^{1/2} dt\).

Solve for \(t^{1/2} dt\) in terms of \(du\): \(t^{1/2} dt = \frac{du}{3}\). Substitute back into the integral to rewrite it entirely in terms of \(u\): the integral becomes \(\int \sin(u) \cdot \frac{du}{3}\).

Integrate with respect to \(u\): \(\frac{1}{3} \int \sin(u) du\). After integrating, substitute back \(u = 2t^{3/2}\) 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 technique 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 method is especially useful when the integral contains a composite function, such as sin(2t^(3/2)).

Handling Powers and Roots in Integrals

Understanding how to manipulate expressions with fractional exponents, like t^(3/2) and √t (which is t^(1/2)), is essential. Converting roots to fractional powers allows easier differentiation and substitution, facilitating the integration process.

Trigonometric Functions in Integration

Integrating functions involving trigonometric expressions, such as sin(2t^(3/2)), requires familiarity with their properties and how they interact with substitution. Recognizing the inner function and its derivative helps in applying substitution effectively.

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