Chapter 5: Q. 56 (page 442)

Calculate each of the integrals in Exercises 53–56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$\int \frac{\ln x + 1}{x {((\ln x)}^{2} - 4)} d x$

Short Answer

Expert verified

The value is $\frac{1}{4} \ln | \ln (x) + 2 | + \frac{3}{4} \ln | \ln (x) - 2 | + C$

Step by step solution

Step 1. Given Information:

Given integral : $\int \frac{\ln x + 1}{x {((\ln x)}^{2} - 4)} d x$

We want to calculate each of the integrals.

Step 2. Calculation:

$U s e s u b s t i t u t i o n \ln (x) + 1 = u A l s o \ln (x) = u - 1 a n d \frac{1}{x} d x = d u S u b s t i t u t i o n v a l u e s i n i n t e g r a l w e g e t = \int \frac{u}{{(u - 1)}^{2} - 4} d u = \int \frac{u}{u^{2} - 2 u - 3} d u U \sin g P a r t i a l F r a c t i o n w e g e t \frac{u}{u^{2} - 2 u - 3} = \frac{1}{4 (u + 1)} + \frac{3}{4 (u - 3)} N o w \int \frac{u}{u^{2} - 2 u - 3} d u = \frac{1}{4} \int \frac{1}{u + 1} d u + \frac{3}{4} \int \frac{1}{u - 3} d u = \frac{1}{4} \ln | u + 1 | + \frac{3}{4} \ln | u - 3 | + C S u b s i t u t e b a c k u = \ln (x) + 1 = \frac{1}{4} \ln | \ln (x) + 1 + 1 | + \frac{3}{4} \ln | \ln (x) + 1 - 3 | + C = \frac{1}{4} \ln | \ln (x) + 2 | + \frac{3}{4} \ln | \ln (x) - 2 | + C$

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Calculate each of the integrals in Exercises 53–56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$\int \frac{\ln x + 1}{x {((\ln x)}^{2} - 4)} d x$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

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Calculate each of the integrals in Exercises 53–56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.∫lnx+1x((lnx)2-4)dx

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Calculation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Mechanics Maths

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Calculate each of the integrals in Exercises 53–56. Each integral requires substitution or integration by parts as well as the algebraic methods described in this section.
$\int \frac{\ln x + 1}{x {((\ln x)}^{2} - 4)} d x$