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:$\int 7x^{2} ln(x) dx \\= ln(x) ( \frac{7x^{3}}{3})-\int( \frac{7x^{3}}{3}) \frac{1}{x} dx \\= \frac{7x^{3}}{3} ln(x)- \frac{7}{3} \int x^{2} dx \\= \frac{7x^{3}}{3}ln(x)- \frac{7x^{3}}{9} +c\\= \frac{7x^{3}}{3}(ln(x)- \frac{1}{3} )+c$

Answer: $\frac{7x^{3}}{3}(ln(x)- \frac{1}{3})+c$

Answer 2

Answer:

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

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]:                                                                $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

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

Step-by-step 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:                                                                     $\displaystyle du = \frac{1}{x} \ dx$
3. Set dv:                                                                                                           $\displaystyle dv = x^2$
4. [dv] Integration Rule [Reverse Power Rule]:                                               $\displaystyle v = \frac{x^3}{3}$

Step 4: Integrate Pt. 3

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

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

Unit: Integration