/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find the integral. (Note: Solve ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 23

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int \arctan x d x $$

Short Answer

Expert verified
The integral \( \int \arctan(x) dx = \arctan(x)*x - \frac{1}{2} \ln |1 + x^2| + c \)

Step by step solution

01

Identify u and dv

To use integration by parts, we first need to choose two parts of the integral, i.e. \( u \) and \( dv \). Here, let's identify \( u = \arctan(x) \) and \( dv = dx \).
02

Compute du and v

Calculate \( du \) and \( v \) using \( du = \frac{1}{1 + x^2} dx \) (as the derivative of \( \arctan(x) = \frac{1}{1+ x^2} \)), and \( v = \int dx = x \).
03

Apply Formula for Integration by parts

The formula for integration by parts is \( \int u dv = u*v - \int v du \). Applying this formula we get: \( \int \arctan(x) dx = \arctan(x)*x - \int x*\frac{1}{1 + x^2} dx \)
04

Simplify the integral

We see that the new integral we need to calculate is simpler: \( \int x*\frac{1}{1 + x^2} dx \). This integral can be solved by substituting the denominator \( 1+x^2 = z \). Therefore \( dz = 2x dx \) or \( dx = \frac{dz}{2x} \). Replacing this in our integral gives us \( \frac{1}{2} \int \frac{dz}{z} = \frac{1}{2} \ln |1 + x^2| + c \). The final answer combines both parts: \( \arctan(x)*x - \frac{1}{2} \ln |1 + x^2| + c \)

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Most popular questions from this chapter

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{1-\sec t}{\cos t-1} d t $$

Consider the integral \(\int_{0}^{\pi / 2} \frac{4}{1+(\tan x)^{n}} d x\) where \(n\) is a positive integer. (a) Is the integral improper? Explain. (b) Use a graphing utility to graph the integrand for \(n=2,4,\) \(8,\) and \(12 .\) (c) Use the graphs to approximate the integral as \(n \rightarrow \infty\). (d) Use a computer algebra system to evaluate the integral for the values of \(n\) in part (b). Make a conjecture about the value of the integral for any positive integer \(n\). Compare your results with your answer in part (c).

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