92–98. Evaluate the following integrals.
92. ∫[1 to √2] y⁸ e^(y²) dy

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Identify the integral to solve: \(\int_{1}^{\sqrt{2}} y^{8} e^{y^{2}} \, dy\).

Consider a substitution to simplify the integral. Let \(u = y^{2}\), which implies \(du = 2y \, dy\) or equivalently \(y \, dy = \frac{du}{2}\).

Rewrite the integral in terms of \(u\). Express \(y^{8}\) as \((y^{2})^{4} = u^{4}\), and replace \(y \, dy\) with \(\frac{du}{2}\). The integral becomes \(\int_{u=1^{2}}^{u=(\sqrt{2})^{2}} u^{4} e^{u} \cdot y^{7} \, dy\). However, since \(y^{7} \, dy\) is not directly replaced by \(du\), we need to express \(y^{8} \, dy\) carefully.

Notice that \(y^{8} \, dy = y^{7} \cdot y \, dy\). Using the substitution, \(y \, dy = \frac{du}{2}\), so \(y^{8} \, dy = y^{7} \cdot \frac{du}{2}\). But \(y^{7} = (y^{2})^{3} \cdot y = u^{3} \cdot y\). This suggests the substitution might be more complex, so consider an alternative approach such as integration by parts.

Set up integration by parts with \(u = y^{7}\) and \(dv = y e^{y^{2}} \, dy\). Then compute \(du\) and \(v\), and apply the integration by parts formula \(\int u \, dv = uv - \int v \, du\). This will help break down the integral into simpler parts.

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