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2 years ago

Represents the sets with a Venn diagram: U = {1,2,3,4,5,6,7,8,9,10, 11, 12} A = {1,2,3,4,6, 12} B = {1,2,4,8}

Mathematics

1 answer:

pychu [463]2 years ago

7 0

Answer:

The required figure is shown below

Step-by-step explanation:

Consider the provided sets:

U = {1,2,3,4,5,6,7,8,9,10,11,12}

A = {1,2,3,4,6, 12}

B = {1,2,4,8}

We need to draw the Venn diagram.

Draw a rectangle which represents set U.

Now, Make circles for set A and set B. Every circle should overlap.

Write the element of each set as shown in figure.

Place elements inside each circle which represents the elements of corresponding sets.

Place common element of set in the section in which all circles shapes overlap.

The required figure is shown below

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Evaluate the following definite integral

mihalych1998 [28]

Answer:

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

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U-Substitution

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