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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Answer from: 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

Answer from: Quest

the correct answer is 5 6 and 7

Answer from: Quest

the correct option is 2.

step-by-step explanation:

according to the given information triangle abc is a right angle triangle with right angle at a. segment ab is 5 and segment ac is 7. point d is on segment bc and angles adb and adc are right angles.

it is given that triangles abd, cad, and cba are similar.

two triangles are similar if their corresponding sides are proportional.

in triangle cba, using pythagoras theorem

$ab^2+ac^2=bc^2$

$ab^2+ac^2=x^2$ )

triangle abd and cba are similar,

$\frac{ab}{bc}=\frac{bd}{ab}$

$ab^2=bc\cdot bd$

$25=bc\cdot bd$ (2)

triangle cad and cba are similar,

$\frac{ac}{bc}=\frac{cd}{ac}$

$ac^2=bc\cdot dc$

$7\times 7=bc\cdot dc$

$49=bc\cdot dc$ )

using (1), (2) and (3), we get

$25+49=x^2$

$74=x^2$

hence proved.

therefore option 2 is correct.

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Use integration by parts to derive the following formula from the table of integrals.
∫x^n . e...

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