Question

Determine whether the statement is true or false, and explain why.

Integration by parts should be used to evaluate ∫^1_0 xe^10x dx.

Answer 1

True

Explanation:

The integral

`\int\limits^1_0 {xe^(10x)} \, dx`

in the integral we have a product of a monomial:
`x`

and an exponential function:
`e^(10x)`

In general case, when we have a combination of these two things you can use the integration by parts, where
`x` will be
`u` and
`e^(10x)` will be
`dv`.

The statement is true

Answer 2

True

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals

Integration Rule [Fundamental Theorem of Calculus 1]:
`\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)`

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

U-Substitution

Integration by Parts:
`\displaystyle \int {u} \, dv = uv - \int {v} \, du`

• [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^1_0 {xe^(10x)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

1. Set u:
   `\displaystyle u = x`
2. [u] Basic Power Rule:
   `\displaystyle du = dx`
3. Set dv:
   `\displaystyle dv = e^(10x) \ dx`
4. [dv] Exponential Integration [U-Substitution]:
   `\displaystyle v = (e^(10x))/(10)`

Step 3: integrate Pt. 2

1. [Integral] Integration by Parts:
   `\displaystyle \int\limits^1_0 {xe^(10x)} \, dx = (xe^(10x))/(10) \bigg| \limits^1_0 - \int\limits^1_0 {(e^(10x))/(10)} \, dx`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^1_0 {xe^(10x)} \, dx = (xe^(10x))/(10) \bigg| \limits^1_0 - (1)/(10) \int\limits^1_0 {e^(10x)} \, dx`
3. [Integral] Exponential Integration:
   `\displaystyle \int\limits^1_0 {xe^(10x)} \, dx = (xe^(10x))/(10) \bigg| \limits^1_0 - (1)/(10) \bigg( (e^(10x))/(10) \bigg) \bigg| \limits^1_0`
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^1_0 {xe^(10x)} \, dx = (9e^(10))/(100) + (1)/(10)`
5. Simplify:
   `\displaystyle \int\limits^1_0 {xe^(10x)} \, dx = (9e^(10) + 1)/(100)`

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

Unit: Integration