   Chapter 5, Problem 33RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Evaluate the integral, if it exists. ∫ tan x ln ( cos x )   d x

To determine

To find: The value of the integral function.

Explanation

Given:

The integral function is tanxln(cosx)dx.

Calculation:

Consider ln(cosx) as f.

f=ln(cosx) (1)

Differentiate both sides of the equation (1).

dfdx=ddx[ln(cosx)]=1cosx(sinx)=sinxcosx (2)

Consider tanx as g then,

g=tanxdgdx=tanxdg=tanxdx (3)

Integrate both sides of the equation (3).

dg=tanxdxg=sinxcosxdx (4)

Consider cosx as u.

u=cosx (5)

Differentiate both sides of the equation.

du=sinxdx (6)

Substitute u for cosx and du for sinxdx in equation (4) as shown below:

g=sinxcosxdx=1u(du)=ln(u) (7)

Substitute cosx for u in equation (7) as shown below:

g=ln(cosx) (8)

Expression to find the integral value is shown below:

tanxln(cosx)dx=sinx(ln(cosx))cosxdxln2(cos(x)) (9)

Consider ln(cosx) as u

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