If you do not know what substitution to make, try reducing

Chapter 5, Problem 67E

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QUESTION:

If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitutions in Exercises 67 and 68.

                       \(\int \frac{18 \tan ^{2} x \sec ^{2} x}{\left(2+\tan ^{3} x\right)^{2}} d x\)

a. \(u=\tan x\), followed by \(v=u^{3}\), then by \(w=2+v\)

b. \(u=\tan ^{3} x\), followed by \(v=2+u\)

c. \(u=2+\tan ^{3} x\)

Equation Transcription:

Text Transcription:

Integral 18 tan^2 x sec^2x/(2+tan^3 x)^2 dx

u=tan x

v=u^3

w=2+v

u=tan^3x

v=2+u

u=2+tan^3x

Questions & Answers

QUESTION:

If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitutions in Exercises 67 and 68.

                       \(\int \frac{18 \tan ^{2} x \sec ^{2} x}{\left(2+\tan ^{3} x\right)^{2}} d x\)

a. \(u=\tan x\), followed by \(v=u^{3}\), then by \(w=2+v\)

b. \(u=\tan ^{3} x\), followed by \(v=2+u\)

c. \(u=2+\tan ^{3} x\)

Equation Transcription:

Text Transcription:

Integral 18 tan^2 x sec^2x/(2+tan^3 x)^2 dx

u=tan x

v=u^3

w=2+v

u=tan^3x

v=2+u

u=2+tan^3x

ANSWER:

Solution

Step 1 of 4

In this problem we have to evaluate the indefinite integral.

Given that

(a) given the substitution


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