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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.2.36
Capitolo 8, Problema 8.2.36

9–40. Integration by parts Evaluate the following integrals using integration by parts.
36. ∫ from 0 to ln2 x eˣ dx

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Identify the integral to solve: \(\int_0^{\ln 2} x e^{x} \, dx\).
Choose functions for integration by parts: let \(u = x\) (which simplifies when differentiated) and $dv = e^{x} dx$ (which is easy to integrate).
Compute the derivatives and integrals needed: $du = dx$ and \(v = \int e^{x} dx = e^{x}\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), so write \(\int_0^{\ln 2} x e^{x} dx = \left. x e^{x} \right|_0^{\ln 2} - \int_0^{\ln 2} e^{x} dx\).
Evaluate the remaining integral \(\int_0^{\ln 2} e^{x} dx\) and then substitute the limits into both terms to express the final answer.

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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 simplifies the integration process.
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06:18
Integration by Parts for Definite Integrals

Definite Integrals

Definite integrals calculate the net area under a curve between two limits. When applying integration by parts to definite integrals, the evaluation of the product uv is done at the upper and lower limits, and the resulting integral is also evaluated within these bounds.
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Definition of the Definite Integral

Exponential and Logarithmic Functions

Understanding the properties of exponential functions (eˣ) and logarithmic functions (ln x) is essential. Their derivatives and integrals are well-known, and recognizing these helps in selecting u and dv in integration by parts, especially when limits involve logarithms.
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Derivatives of General Logarithmic Functions
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