   Chapter 8.5, Problem 32E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Integrating a Trigonometric Function In Exercises 1–32, find the indefinite integral. See Examples 1, 2, 3, 4, 5, and 8. ∫ ( 1 + tan θ ) 2 d θ

To determine

To calculate: The integral of (1+tanθ)2dθ.

Explanation

Given Information:

The provided integral is (1+tanθ)2dθ.

Formula used:

Write the formula for (a+b)2.

(a+b)2=a2+b2+2ab

Write the formula for integral of sec2u.

sec2udu=tanu+C

Write the formula for integral of tanu.

tanudu=ln|cosu|+C

Write the formula for trigonometrical identity.

sec2u=tan2u+1,

Calculation:

Consider the provided integral,

(1+tanθ)2dθ

Use the formulae of trigonometrical identity and (a+b)2 to simplify the provided integral.

(1+tanθ)2dθ=(1+tan2θ+2tanθ)dθ=(sec2θ+2tanθ)dθ=sec2θdθ+2tanθdθ

Let u=θ

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