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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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)).

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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.

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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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