Question

Please help if you know how to integrate using substitution.

Answer 1

B.
`\displaystyle (-1)/(2)ln|cos2x| + C`

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Calculus

Antiderivatives - Integrals

Integral Constant C

U-Substitution

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

Trig Integration:
`\displaystyle \int {tan(u)} \, du = -ln|cosu| + C`

Explanation:

Step 1: Define

`\displaystyle \int {tan(2x)} \, dx`

Step 2: Identify Variables

Set variables for u-substitution.

u = 2x

du = 2dx

Step 3: Integrate

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle (1)/(2)\int {2tan(2x)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle (1)/(2)\int {tanu} \, du`
3. [Integral] Trig Integration:
   `\displaystyle (1)/(2)(-ln|cosu|) + C`
4. [Integral] Back-Substitute:
   `\displaystyle (-1)/(2)ln|cos2x| + C`

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

Unit: Integration

Book: College Calculus 10e