   Chapter 8.5, Problem 29E

Chapter
Section
Textbook Problem

# Finding an Indefinite Integral In Exercises 25-32, use substitution and partial fraction to find the indefinite integral. ∫ sec 2 x tan 2 x + 5 tan x + 6 d x .

To determine
The solution of the given indefinite integral by using partial fractions.

Explanation

Given:

The indefinite integral is sec2xtan2x+5tanx+6dx.

Calculation:

Consider the following indefinite integral,

sec2xtan2x+5tanx+6dx. …...…..... (1)

Substitute u=tanx and differentiate both sides,

du=sec2xdx

Substitute the value in equation (1),

sec2xtan2x+5tanx+6dx=duu2+5u+6

We can write,

1u2+5u+6=1(u+3)(u+2)

Now by using partial fraction method we get,

1(u+3)(u+2)=Au+3+Bu+2

By simplifying further we get,

1=A(u+2)+B(u+3)

At u=-3,

1=A(3+2)+B(3+3)A=11A=1

At u=-2,

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