Question

find each of the following indefinite integrals.


∫sec^2 θ √ tanθ dθ

Answer 1

Answer:

$\displaystyle \int {sec^2(\theta)\sqrt{tan(\theta)}} \, d\theta = \frac{2[tan(\theta)]^\bigg{\frac{3}{2}}}{3} + C$

General Formulas and Concepts:

Algebra I

• Exponential Rule [Root Rewrite]:                                                                 $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$

Calculus

Differentiation

• Derivatives
• Derivative Notation

Integration

• Integrals
• Indefinite Integrals
• Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int {sec^2(\theta)\sqrt{tan(\theta)}} \, d\theta$

Step 2: Integrate Pt. 1

1. [Integrand] Rewrite [Exponential Rule - Root Rewrite]:                               $\displaystyle \int {sec^2(\theta)\sqrt{tan(\theta)}} \, d\theta = \int {sec^2(\theta)[tan(\theta)]^\bigg{\frac{1}{2}}} \, dx$

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

1. Set u:                                                                                                             $\displaystyle u = tan(\theta)$
2. [u] Differentiate [Trigonometric Differentiation]:                                         $\displaystyle du = sec^2(\theta) \ d\theta$

Step 4: Integrate Pt. 3

1. [Integral] U-Substitution:                                                                               $\displaystyle \int {sec^2(\theta)\sqrt{tan(\theta)}} \, d\theta = \int {u^\bigg{\frac{1}{2}}} \, du$
2. [Integral] Reverse Power Rule:                                                                     $\displaystyle \int {sec^2(\theta)\sqrt{tan(\theta)}} \, d\theta = \frac{2u^\bigg{\frac{3}{2}}}{3} + C$
3. Back-Substitute:                                                                                             $\displaystyle \int {sec^2(\theta)\sqrt{tan(\theta)}} \, d\theta = \frac{2[tan(\theta)]^\bigg{\frac{3}{2}}}{3} + C$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e