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Chapter 5: Problem 35

Evaluate using integration by parts. $$ \int_{0}^{1} x e^{x} d x $$

Short Answer

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Step by step solution

01

- Identify parts for integration by parts

In integration by parts, use the formula: \[ \text{鈭� u dv = uv - 鈭� v du} \] Let \( u = x \) and \( dv = e^x dx \).
02

- Differentiate and integrate

Differentiate \( u \) and integrate \( dv \): \( u = x \rightarrow du = dx \)\( dv = e^x dx \rightarrow v = e^x \)
03

- Apply the integration by parts formula

Substitute \( u, v, du, \) and \( dv \) into the formula: \[ \text{鈭� x e^x dx = x e^x - 鈭� e^x dx} \]
04

- Integrate \( 鈭� e^x dx \)

Integrate \( e^x \):\[ \text{鈭� e^x dx = e^x} \]
05

- Substitute back and evaluate the definite integral

Substitute back and evaluate from 0 to 1:\[ \text{x e^x - e^x \bigg|_{0}^{1} = \left( 1 \times e^{1} - e^{1} \right) - \left( 0 \times e^{0} - e^{0} \right)} \]\[ \text{= \left( e - e \right) - \left( 0 - 1 \right) = -1} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integral
A definite integral represents the area under a curve, from one point to another. In mathematical terms, this is written as \(\text{鈭玙{a}^{b} f(x) dx}\), where \(a\) and \(b\) are the bounds of integration. For our problem, the integral \(\text{鈭玙{0}^{1} x e^{x} dx}\) represents the area under the curve \(x e^x\) from \(x = 0\) to \(x = 1\). To find this, we use integration by parts, it helps to break down complex integrals.
Integration Techniques
When solving integrals, especially complex ones, using different techniques can simplify the process. One such method is Integration by Parts, useful for products of functions. The formula for integration by parts is: \[ \text{鈭� u dv = uv - 鈭� v du} \] This technique requires identifying parts of the integrand (the function to be integrated). For \(\text{鈭玙{0}^{1} x e^{x} dx}\):\
  • Let \(u = x\) (differentiate to get \(du = dx\))
  • Let \(dv = e^x dx\) (integrate to get \(v = e^x\))
Substituting these into the formula produces a simpler expression to integrate.
Calculus for Life Sciences
Understanding calculus, especially integration, is crucial in life sciences. It's used in modeling population growth, the spread of diseases, and even drug efficiency. Definite integrals, for instance, can be used to calculate the total production of a protein over a period. In our problem, integrating \(x e^x\) could represent a biological process with a rate proportional to both x and exponential growth over time. Techniques like integration by parts help solve these real-world problems by breaking down complex integrals into simpler, more manageable parts.

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Most popular questions from this chapter

Evaluate using integration by parts. Check by differentiating. $$ \int x \csc ^{2} x d x $$

Evaluate using integration by parts. Check by differentiating. $$ \int x e^{-x} d x $$

Evaluate using integration by parts. Check by differentiating. $$ \int \ln (x+3) d x $$

Evaluate using integration by parts. \(\int_{1}^{2} x^{2} \ln x d x\)

a) Integrate \(\int x \sqrt[3]{x+2} d x\) using a substitution. b) Use integration by parts to compute \(\int x \sqrt[3]{x+2} d x\) c) Use algebra to show that your answers in parts (a) and (b) are equal.
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