Question

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

Accepted Answer

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.

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

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주요 개념

Integration by Substitution

Exponential Functions in Integration

Definite Integrals and Limits of Integration