Question

Integrate ∫2tan(2x+π)dx​

Answer 1

Answer:

$\displaystyle \int {2 \tan (2x + \pi)} \, dx = - \ln \big| \cos (2x + \pi) \big| + C$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• Indefinite Integrals
• Integration Constant C

Integration Property [Multiplied Constant]:                                                         $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

Trigonometric Integration

Step-by-step explanation:

Step 1: Define

Identify.

$\displaystyle \int {2 \tan (2x + \pi)} \, dx$

Step 2: Integrate Pt. 1

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int {2 \tan (2x + \pi)} \, dx = 2 \int {\tan (2x + \pi)} \, dx$

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

1. Set u:                                                                                                             $\displaystyle u = 2x + \pi$
2. [u] Differentiate [Derivative Properties]:                                                       $\displaystyle du = 2 \ dx$

Step 4: Integrate Pt. 3

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int {2 \tan (2x + \pi)} \, dx = \frac{2}{2} \int {2 \tan (2x + \pi)} \, dx$
2. [Integral] U-Substitution:                                                                               $\displaystyle \int {2 \tan (2x + \pi)} \, dx = \frac{2}{2} \int {\tan u} \, du$
3. Simplify:                                                                                                         $\displaystyle \int {2 \tan (2x + \pi)} \, dx = \int {\tan u} \, du$
4. [Integral] Trigonometric Substitution:                                                           $\displaystyle \int {2 \tan (2x + \pi)} \, dx = - \ln \big| \cos u \big| + C$
5. [u] Back-Substitute:                                                                                       $\displaystyle \int {2 \tan (2x + \pi)} \, dx = - \ln \big| \cos (2x + \pi) \big| + C$

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

Unit: Integration