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3 years ago

A zoo has some new exhibits.

Mathematics

1 answer:

Licemer1 [7]3 years ago

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Answer

Elephant: 14%

Panda: 18%

Tiger: 29%

Bonobo: 39%:

Step-by-step explanation:

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Evaluate the following definite integral

mihalych1998 [28]

Answer:

$\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}$

General Formulas and Concepts:

Symbols

e (Euler's number) ≈ 2.71828

Algebra I

Exponential Rule [Multiplying]: $\displaystyle b^m \cdot b^n = b^{m + n}$

Calculus

Differentiation

Derivatives
Derivative Notation

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Integration

Integrals
Definite Integrals
Integration Constant C

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

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

U-Solve

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

$\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx$

Step 2: Integrate Pt. 1

[Integrand] Rewrite [Exponential Rule - Multiplying]: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \int\limits^1_0 {x^5e^{x^3}e} \, dx$
[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = e\int\limits^1_0 {x^5e^{x^3}} \, dx$

Step 3: Integrate Pt. 2

Identify variables for u-solve.

Set u: $\displaystyle u = x^3$
[u] Differentiate [Basic Power Rule]: $\displaystyle du = 3x^2 \ dx$
[u] Rewrite: $\displaystyle x = \sqrt[3]{u}$
[du] Rewrite: $\displaystyle dx = \frac{1}{3x^2} \ du$

Step 4: Integrate Pt. 3

[Integral] U-Solve: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = e\int\limits^1_0 {x^5e^{(\sqrt[3]{u})^3}\frac{1}{3x^2}} \, du$
[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}\int\limits^1_0 {x^5e^{(\sqrt[3]{u})^3}\frac{1}{x^2}} \, du$
[Integral] Simplify: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}\int\limits^1_0 {x^3e^u} \, du$
[Integrand] U-Solve: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}\int\limits^1_0 {ue^u} \, du$

Step 5: integrate Pt. 4

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = u$
[u] Differentiate [Basic Power Rule]: $\displaystyle du = du$
Set dv: $\displaystyle dv = e^u \ du$
[dv] Exponential Integration: $\displaystyle v = e^u$

Step 6: Integrate Pt. 5

[Integral] Integration by Parts: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3} \bigg[ ue^u \bigg| \limits^1_0 - \int\limits^1_0 {e^u} \, du \bigg]$
[Integral] Exponential Integration: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3} \bigg[ ue^u \bigg| \limits^1_0 - e^u \bigg| \limits^1_0 \bigg]$
Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}[ e - e ]$
Simplify: $\displaystyle \int\limits^1_0 {x^5e^{x^3 + 1}} \, dx = \frac{e}{3}$