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Chapter 7: Problem 5

What type of integrand is a good candidate for integration by parts?

Short Answer

Expert verified
Question: Explain the process of determining if an integrand is suitable for integration by parts and list the steps required to solve it. Answer: To determine if an integrand is suitable for integration by parts, we need to identify if there are two functions within the integrand, one of which can be easily differentiated and the other easily integrated. Ideally, differentiating one of the functions should simplify the expression, while integrating the other should not make it significantly more complicated. The steps to solve an integral using integration by parts are: 1. Identify the two functions in the integrand that are easy to differentiate and integrate. 2. Differentiate the function that simplifies when differentiated, and integrate the function that does not become more complicated when integrated. 3. Apply the integration by parts formula: \[ \int u\ dv = uv - \int v\ du \] where u is the function that is being differentiated, and dv is the function that is being integrated. Plug in the differentiated and integrated terms of both identified functions and solve the integral.

Step by step solution

01

Identify the two functions within the integrand

First, look for an integrand that can be broken down into a product of two functions. One of these functions should be easy to differentiate, and the other should be easy to integrate.
02

Differentiate and integrate the functions

Differentiate the function that simplifies when differentiated, and integrate the function that does not become more complicated when integrated.
03

Apply integration by parts formula

Integration by parts follows the formula: \[ \int u\ dv = uv - \int v\ du \] where u is the function that is being differentiated, and dv is the function that is being integrated. Plug in the differentiated and integrated terms of both identified functions and solve the integral.

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Most popular questions from this chapter

Many methods needed Show that \(\int_{0}^{\infty} \frac{\sqrt{x} \ln x}{(1+x)^{2}} d x=\pi\) in the following steps. a. Integrate by parts with \(u=\sqrt{x} \ln x.\) b. Change variables by letting \(y=1 / x.\) c. Show that \(\int_{0}^{1} \frac{\ln x}{\sqrt{x}(1+x)} d x=-\int_{1}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} d x\) and conclude that \(\int_{0}^{\infty} \frac{\ln x}{\sqrt{x}(1+x)} d x=0.\) d. Evaluate the remaining integral using the change of variables \(z=\sqrt{x}\) (Source: Mathematics Magazine 59, No. 1 (February 1986): 49).

On the interval \([0,2],\) the graphs of \(f(x)=x^{2} / 3\) and \(g(x)=x^{2}\left(9-x^{2}\right)^{-1 / 2}\) have similar shapes. a. Find the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval [0,2] b. Find the area of the region bounded by the graph of \(g\) and the \(x\) -axis on the interval [0,2] c. Which region has the greater area?

Use integration by parts to evaluate the following integrals. $$\int_{0}^{\infty} x e^{-x} d x$$

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{x \sqrt{1+2 x}} ; 1+2 x=u^{2}$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{\left(e^{x}+e^{-x}\right)^{2}}$$
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