Evaluate the integrals in Exercises 87–96.
91. ∫₁^(√2) x·2^(x²) dx

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Identify the integral to solve: $\int_{1}^{\sqrt{2}} x \cdot 2^{x^{2}} \, dx$.

Recognize that the integrand contains the function \$2^{x^{2}}$ multiplied by $x$, and notice that the exponent $x^{2}$ suggests a substitution involving $x^{2}$.

Use the substitution method: let $u = x^{2}$. Then, compute the differential $du = 2x \, dx$, which implies $x \, dx = \frac{du}{2}$.

Change the limits of integration according to the substitution: when $x = 1$, $u = 1^{2} = 1$; when $x = \sqrt{2}$, $u = (\sqrt{2})^{2} = 2$.

Rewrite the integral in terms of $u$: $\int_{1}^{2} 2^{u} \cdot \frac{1}{2} \, du = \frac{1}{2} \int_{1}^{2} 2^{u} \, du$. Then, proceed to integrate \$2^{u}$ with respect to $u$.

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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 an exponential with a function inside.

Exponential Functions with Variable Exponents

Exponential functions with variable exponents, like 2^(x²), involve raising a constant base to a function of x. Understanding how to differentiate and integrate such functions requires applying the chain rule or substitution, as the exponent itself is a function rather than a constant.

Definite Integrals and Limits of Integration

Definite integrals calculate the net area under a curve between two specific points, called limits of integration. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus to compute the difference at the upper and lower limits.

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