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Chapter 5: Q 13. (page 495)

Fill in the blanks to complete each of the following integration formulas.
$\int \ln x d x =_________$

Short Answer

Expert verified

The result is $x \ln (x) - x + C$ .

Step by step solution

01

Step 1. Given information.

Consider the given integration.

$\int \ln x d x$

02

Step 2. Find the integration.

Apply the by-parts integration method. Consider the values as $u = 1$ and $v = \ln x$ .

$\begin{array}{rcl} \int \ln x d x & = & x \ln x - \int x (\frac{1}{x}) d x \\ = & x \ln x - \int d x \\ = & x \ln x - x + C \end{array}$

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Most popular questions from this chapter

Solve the integral: $\int x \ln \sqrt{x} d x$

Suppose v(x) is a function of x. Explain why the integral

of dv is equal to v (up to a constant).

Consider the integral $\int \frac{1}{x^{2} \sqrt{1 - x^{2}}} d x$ from the reading at the beginning of the section.

(a) Use the inverse trigonometric substitution $u = \sin^{- 1} x$ to solve this integral.

(b) Use the trigonometric substitution $x = \sin u$ to solve the integral.

(c) Compare and contrast the two methods used in parts (a) and (b).

Solve $\int \frac{1}{x^{2} + 9} d x$ the following two ways:

(a) with the trigonometric substitution x = 3 tan u;

(b) with algebra and the derivative of the arctangent.

Explain why using trigonometric substitution with $x = a \tan u$ often involves a triangle with side lengths a and x and hypotenuse of length $\sqrt{x^{2} + a^{2}}$

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