Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
19. ∫ 1/√(x² - 81) dx, x > 9
Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.2.36
9–40. Integration by parts Evaluate the following integrals using integration by parts.
36. ∫ from 0 to ln2 x eˣ dx
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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05:43Definition 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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06:32Derivatives of General Logarithmic Functions
Related Practice
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Textbook Question
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Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
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Textbook Question
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Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
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