Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt

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Identify the integral to solve: $\int 2t e^{-t^{2}} \, dt$.

Recognize that the integrand contains a function and its derivative: the exponent $-t^{2}$ and the factor \$2t$ which is the derivative of $-t^{2}$ up to a constant.

Use substitution by letting $u = -t^{2}$. Then compute the differential: $du = -2t \, dt$, which implies $-du = 2t \, dt$.

Rewrite the integral in terms of $u$: $\int 2t e^{-t^{2}} \, dt = \int e^{u} (-du) = -\int e^{u} \, du$.

Integrate with respect to $u$: $-\int e^{u} \, du = -e^{u} + C$, then substitute back $u = -t^{2}$ to get the final expression in terms of $t$.

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주요 개념

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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, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a function and its derivative.

Exponential Functions in Integration

Exponential functions, such as e^(x), often appear in integrals and require special attention. Understanding how to integrate expressions involving e raised to a function is crucial, especially when combined with polynomial terms. Recognizing patterns helps in applying substitution effectively.

Differentiation of Composite Functions

Differentiation of composite functions involves applying the chain rule to find derivatives of functions within functions. This concept is important in integration when reversing the differentiation process, as it helps identify suitable substitutions by recognizing inner functions and their derivatives.

Evaluate the integrals in Exercises 33–54.∫ 2t e^(-t²) dt