Question

Evaluate the integral using integration by parts with the indicated choices of u and dv. (use c for the constant of integration.) 7x2 ln(x) dx; u

Answer 1

Answer:
`\int 7x^(2) ln(x) dx \\= ln(x) ( (7x^(3))/(3))-\int( (7x^(3))/(3)) (1)/(x) dx \\= (7x^(3))/(3) ln(x)- (7)/(3) \int x^(2) dx \\= (7x^(3))/(3)ln(x)- (7x^(3))/(9) +c\\= (7x^(3))/(3)(ln(x)- (1)/(3) )+c`

Answer:
`(7x^(3))/(3)(ln(x)- (1)/(3))+c`

Answer 2

`\displaystyle \int {7x^2 \ln x} \, dx = (7x^3)/(3) \bigg( \ln(x) - (1)/(3) \bigg) + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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`

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

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 {7x^2 \ln x} \, dx`

Step 2: Integrate Pt. 1

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {7x^2 \ln x} \, dx = 7 \int {x^2 \ln x} \, dx`

Step 3: Integrate Pt. 2

Identify variables for integration by parts using LIPET.

1. Set u:
   `\displaystyle u = \ln x`
2. [u] Logarithmic Differentiation:
   `\displaystyle du = (1)/(x) \ dx`
3. Set dv:
   `\displaystyle dv = x^2`
4. [dv] Integration Rule [Reverse Power Rule]:
   `\displaystyle v = (x^3)/(3)`

Step 4: Integrate Pt. 3

1. [Integral] Integration by Parts:
   `\displaystyle \int {7x^2 \ln x} \, dx = 7 \bigg( (x^3 \ln(x))/(3) - \int {(x^2)/(3)} \, dx \bigg)`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {7x^2 \ln x} \, dx = 7 \bigg( (x^3 \ln(x))/(3) - (1)/(3) \int {x^2} \, dx \bigg)`
3. Factor:
   `\displaystyle \int {7x^2 \ln x} \, dx = (7)/(3) \bigg( x^3 \ln(x) - \int {x^2} \, dx \bigg)`
4. [Integral] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int {7x^2 \ln x} \, dx = (7)/(3) \bigg( x^3 \ln(x) - (x^3)/(3) \bigg) + C`
5. Factor:
   `\displaystyle \int {7x^2 \ln x} \, dx = (7x^3)/(3) \bigg( \ln(x) - (1)/(3) \bigg) + C`

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

Unit: Integration