Evaluate each iterated integral. (See the indicated problem...

Evaluate each iterated integral. (See the indicated problem for the evaluation of the inner integral.)
$$
\begin{aligned}
&\int_{0}^{2} \int_{1}^{5} \frac{x}{\sqrt{y+x^{2}}} d y d x\\
&\text { (See Problem 12.) }
\end{aligned}
$$

Key Concepts

Iterated Integrals

An iterated integral is the repeated application of integration in multiple variables, typically used to compute double or triple integrals by integrating one variable at a time. This method simplifies multidimensional integration by breaking it down into a series of single-variable integrals, often determined by the limits of integration provided in a prescribed order.

Inner Integration

The inner integration refers to the process of integrating with respect to one variable while treating the other variables as constants. It is the first step in evaluating an iterated integral and is crucial because its result becomes a function of the remaining variables, which are then integrated in the next step.

Substitution Method

The substitution method, also known as u-substitution, is a technique used to simplify the integration of composite functions. It involves changing variables to transform the integrand into a form that is easier to integrate. This method is especially useful when the integrand contains a function and its derivative, allowing the integral to be evaluated more straightforwardly.

Definite Integration

Definite integration involves evaluating an integral over a specific interval, producing a numerical result that represents the area under a curve or the accumulated quantity between the given bounds. In this context, it requires evaluating the antiderivative at the endpoints of the interval and taking their difference, and it is used in both the inner and outer integrations of iterated integrals.

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