Mathematics, 14.01.2020 22:31, lexie7752

Use integration by parts to derive the following formula from the table of integrals.
∫x^n . e^ax dx=x^ne^ax/a-n/a∫x^n-1e^ax dx +c, a≠-0.

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answered: testedagent2823

See explanation.

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals[Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

Step-by-step explanation:

*Note:

Treat a and n as arbitrary constants.

Step 1: Define

Identify

$\displaystyle \int {x^ne^{ax}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = x^n$ [u] Basic Power Rule: $\displaystyle du = nx^{n - 1} \ dx$ Set dv: $\displaystyle dv = e^{ax} \ dx$ [dv] Exponential Integration [U-Substitution]: $\displaystyle v = \frac{e^{ax}}{a}$

Step 3: Integrate Pt. 2

[Integral] Integration by Parts: $\displaystyle \int {x^ne^{ax}} \, dx = \frac{x^na^{ax}}{a} - \int {\frac{nx^{n - 1}e^{ax}}{a}} \, dx$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {x^ne^{ax}} \, dx = \frac{x^na^{ax}}{a} - \frac{n}{a} \int {x^{n - 1}e^{ax}} \, dx ,\ a \neq 0$

∴ We have verified/derived the formula.

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

Unit: Integration

answered: Guest

The answer would be -3.9> -4.5

answered: Guest

Answer to your question: a. the fundamental theorem of algebra tells you that the equation will have two complex roots since the degree of the polynomial is 2. the roots are x=1±i7√2 .number of roots=degree of polynomialhere, degree of polynomial is 2 so number of roots will be 2.hope it *_*