9–40. Integration by parts Evaluate the following integrals using integration by parts.
40. ∫ e^√x dx

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Identify the integral to solve: \(\int e^{\sqrt{x}} \, dx\).

Use substitution to simplify the integral. Let \(t = \sqrt{x}\), which implies \(x = t^2\). Then, differentiate to find \(dx\): \(dx = 2t \, dt\).

Rewrite the integral in terms of \(t\): \(\int e^{t} \, dx = \int e^{t} \cdot 2t \, dt = 2 \int t e^{t} \, dt\).

Apply integration by parts to \(\int t e^{t} \, dt\). Choose \(u = t\) (so \(du = dt\)) and \(dv = e^{t} dt\) (so \(v = e^{t}\)). Use the formula \(\int u \, dv = uv - \int v \, du\).

Substitute back the expressions for \(u\), \(v\), and \(du\) into the integration by parts formula to express the integral, then multiply by 2 as per the substitution step. Finally, replace \(t\) with \(\sqrt{x}\) to return to the original variable.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely is crucial to simplify the integral effectively.

Substitution Method

The substitution method involves changing variables to simplify an integral. For integrals like ∫e^√x dx, substituting t = √x helps rewrite the integral in terms of t, making it easier to apply integration techniques such as integration by parts.

Handling Composite Functions in Integration

Integrating functions like e^√x requires understanding how to deal with composite functions, where one function is nested inside another. Recognizing the inner function and its derivative is essential for applying substitution or integration by parts correctly.

9–40. Integration by parts Evaluate the following integrals using integration by parts.40. ∫ e^√x dx