Mathematics High School

Hello Calculus!

Find the value

[tex]\\ \rm\Rrightarrow {\displaystyle{\int\limits_3^5}}(e^{3x}+7cosx-3tan^3x)dx[/tex]

Note:-

Answer with proper explanation required and all steps to be mentioned .

​

Answers

Answer 1

Answer:

[tex]\displaystyle \int\limits^5_3 {\bigg( e^{3x} + 7 \cos x - 3 \tan^3 x \bigg)} \, dx = \frac{e^{15} - e^9}{3} + 7 \bigg( \sin 5 - \sin 3 \bigg) - 3 \bigg( \frac{\sec^2 5 - \sec^2 3}{2} - \ln \bigg| \frac{\cos 3}{\cos 5} \bigg| \bigg)[/tex]

General Formulas and Concepts:
Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]

Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]

Derivative Rule [Basic Power Rule]:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals

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

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Method: U-Substitution + U-Solve

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \int\limits^5_3 {\bigg( e^{3x} + 7 \cos x - 3 \tan^3 x \bigg)} \, dx[/tex]

Step 2: Integrate Pt. 1

[Integral] Rewrite [Integration Rule - Addition/Subtraction]:
[tex]\displaystyle \int\limits^5_3 {\bigg( e^{3x} + 7 \cos x - 3 \tan^3 x \bigg)} \, dx = \int\limits^5_3 {e^{3x}} \, dx + \int\limits^5_3 {7 \cos x} \, dx - \int\limits^5_3 {3 \tan^3 x} \, dx[/tex][Integrals] Rewrite [Integration Property - Multiplied Constant]:
[tex]\displaystyle \int\limits^5_3 {\bigg( e^{3x} + 7 \cos x - 3 \tan^3 x \bigg)} \, dx = \int\limits^5_3 {e^{3x}} \, dx + 7 \int\limits^5_3 {\cos x} \, dx - 3 \int\limits^5_3 {\tan^3 x} \, dx[/tex][3rd Integral] Rewrite:
[tex]\displaystyle \int\limits^5_3 {\bigg( e^{3x} + 7 \cos x - 3 \tan^3 x \bigg)} \, dx = \int\limits^5_3 {e^{3x}} \, dx + 7 \int\limits^5_3 {\cos x} \, dx - 3 \int\limits^5_3 {\tan^2 x \bigg( \sec^2 x - 1 \bigg)} \, dx[/tex]

Step 3: Integrate Pt. 2

Identify variables for u-substitution and u-solve.

1st Integral

Set u:
[tex]\displaystyle u = 3x[/tex][u] Differentiate [Derivative Properties and Rules]:
[tex]\displaystyle du = 3 \, dx[/tex][Bounds] Swap:
[tex]\displaystyle \left \{ {{x = 5 \rightarrow u = 3(5) = 15} \atop {x = 3 \rightarrow u = 3(3) = 9}} \right.[/tex]

3rd Integral

Set v:
[tex]\displaystyle v = \sec x[/tex][v] Differentiate [Trigonometric Differentiation]:
[tex]\displaystyle dv = \sec x \tan x \, dx[/tex][dv] Rewrite:
[tex]\displaystyle dx = \frac{1}{\sec x \tan x} \, dv[/tex]

Step 4: Integrate Pt. 3

Let's focus on the 3rd integral first.

Apply Integration Method [U-Solve]:
[tex]\displaystyle 3 \int\limits^5_3 {\tan^2 x \bigg( \sec^2 x - 1 \bigg)} \, dx = 3 \int\limits^{x = 5}_{x = 3} {\frac{v^2 - 1}{v}} \, dv[/tex][Integral] Rewrite [Integration Property - Addition/Subtraction]:
[tex]\displaystyle 3 \int\limits^5_3 {\tan^2 x \bigg( \sec^2 x - 1 \bigg)} \, dx = 3 \Bigg( \int\limits^{x = 5}_{x = 3} {v} \, dv - \int\limits^{x = 5}_{x = 3} {\frac{1}{v}} \, dv \Bigg)[/tex][Integrals] Apply Integration Rules [Reverse Power Rule and Logarithmic Integration]:
[tex]\displaystyle 3 \int\limits^5_3 {\tan^2 x \bigg( \sec^2 x - 1 \bigg)} \, dx = 3 \Bigg( \frac{v^2}{2} \bigg| \limits^{x = 5}_{x = 3} - \ln | v | \bigg| \limits^{x = 5}_{x = 3} \Bigg)[/tex][v] Back-Substitute:
[tex]\displaystyle 3 \int\limits^5_3 {\tan^2 x \bigg( \sec^2 x - 1 \bigg)} \, dx = 3 \Bigg( \frac{\sec^2 x}{2} \bigg| \limits^{5}_{3} - \ln | \sec x | \bigg| \limits^{5}_{3} \Bigg)[/tex]Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
[tex]\displaystyle 3 \int\limits^5_3 {\tan^2 x \bigg( \sec^2 x - 1 \bigg)} \, dx = 3 \Bigg( \frac{\sec^2 5 - \sec^2 3}{2}- \ln \bigg| \frac{\cos 3}{\cos 5} \bigg| \Bigg)[/tex]

Step 5: Integrate Pt. 4

Focus on the other 2 integrals and solve using integration techniques listed above.

1st Integral:

[tex]\displaystyle\begin{aligned}\int\limits^5_3 {e^{3x}} \, dx & = \frac{1}{3} \int\limits^5_3 {3e^{3x}} \, dx \\& = \frac{1}{3} \int\limits^{15}_9 {e^{u}} \, du \\& = \frac{1}{3} e^u \bigg| \limits^{15}_9 \\& = \frac{1}{3} \bigg( e^{15} - e^9 \bigg)\end{aligned}[/tex]

2nd Integral:
[tex]\displaystyle\begin{aligned}7 \int\limits^5_3 {\cos x} \, dx & = 7 \sin x \bigg| \limits^5_3 \\& = 7 \bigg( \sin 5 - \sin 3 \bigg)\end{aligned}[/tex]

Step 6: Integrate Pt. 5

[Integrals] Substitute in integrals:
[tex]\displaystyle \int\limits^5_3 {\bigg( e^{3x} + 7 \cos x - 3 \tan^3 x \bigg)} \, dx = \frac{e^{15} - e^9}{3} + 7 \bigg( \sin 5 - \sin 3 \bigg) - 3 \bigg( \frac{\sec^2 5 - \sec^2 3}{2} - \ln \bigg| \frac{\cos 3}{\cos 5} \bigg| \bigg)[/tex]

∴ we have evaluated the integral.

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Learn more about integration: https://brainly.com/question/27026907

Learn more about calculus: https://brainly.com/question/20158587

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

Unit: Integration

Answer 2

Answer:

1086950.36760

Formula's used:

[tex]\rightarrow \sf \int sin(ax+b)=-\dfrac{1}{a} cos(ax+b)+c[/tex]

[tex]\rightarrow \sf \int cos(ax+b)=\dfrac{1}{a} sin(ax+b)+c[/tex]

[tex]\rightarrow \sf \int \dfrac{1}{ax+b} =\dfrac{1}{a} ln|ax+b|+c[/tex]

[tex]\rightarrow \sf \int e^{ax+b}=\dfrac{1}{a} e^{ax+b} + c[/tex]

[tex]\rightarrow \bold{ ln|a| - ln|b| = ln|\frac{a}{b} | }[/tex]

Explanation:

[tex]\dashrightarrow \sf \int \left(e^{3x}+7cos\left(x\right)-3tan^3\left(x\right)\right)[/tex]

                        apply sum rule: [tex]\bold{\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx}[/tex]

[tex]\dashrightarrow \sf \int \:e^{3x}dx+\int \:7\cos \left(x\right)dx-\int \:3\tan ^3\left(x\right)dx[/tex]

                Integrate simple followings first, using formula's given above

[tex]\dashrightarrow \sf \dfrac{1}{3}e^{3x}+7\sin \left(x\right)-\int 3tan^3x[/tex]

                        Breakdown the component

[tex]\dashrightarrow \sf \dfrac{1}{3}e^{3x}+7\sin \left(x\right)-3\int tan^2x(tanx)[/tex]

                                                         [ tan²x = sec²x - 1 ]

[tex]\dashrightarrow \sf \dfrac{1}{3}e^{3x}+7\sin \left(x\right)-3\int (sec^2x-1)(tanx)[/tex]

===========================================================

for integration of [tex]\bold{\int (sec^2x-1)(tanx)}[/tex]

                                                  apply substitution ... u

[tex]\dashrightarrow \int \dfrac{-1+u^2}{u}[/tex]

[tex]\dashrightarrow \sf \int \:-\dfrac{1}{u}+udu[/tex]

[tex]\dashrightarrow \sf - \int \dfrac{1}{u}du+\int \:udu[/tex]

[tex]\dashrightarrow -\ln \left|u\right|+\dfrac{u^2}{2}[/tex]

substitute back u = sec(x)

[tex]\dashrightarrow \sf-\ln \left|\sec \left(x\right)\right|+\dfrac{\sec ^2\left(x\right)}{2}[/tex]

================================================= insert back

[tex]\dashrightarrow \sf \dfrac{1}{3}e^{3x}+7\sin \left(x\right)-3\left(-\ln \left|\sec \left(x\right)\right|+\dfrac{\sec ^2\left(x\right)}{2}\right)[/tex]   outcome after integrating

Now apply the given limits

[tex]\sf \hookrightarrow \sf \dfrac{1}{3}e^{3(5)}+7\sin \left(5\right)-3\left(-\ln \left|\sec \left(5\right)\right|+\dfrac{\sec ^2\left(5\right)}{2}\right) - (\sf \dfrac{1}{3}e^{3(3)}+7\sin \left(3\right)-3\left(-\ln \left|\sec \left(3\right)\right|+\dfrac{\sec ^2\left(3\right)}{2}\right))[/tex]

                                                                   simplify

[tex]\sf \hookrightarrow \sf \dfrac{1}{3}e^{15}+7\sin \left(5\right)-3\left(-\ln \left|\sec \left(5\right)\right|+\dfrac{\sec ^2\left(5\right)}{2}\right) - (\sf \dfrac{1}{3}e^{9}+7\sin \left(3\right)-3\left(-\ln \left|\sec \left(3\right)\right|+\dfrac{\sec ^2\left(3\right)}{2}\right))[/tex]

                          and group the variables

[tex]\sf \hookrightarrow \dfrac{e^{15}-e^9}{3}-\dfrac{3}{2\cos ^2\left(5\right)}+\dfrac{3}{2\cos ^2\left(3\right)}+7\sin \left(5\right)-7\sin \left(3\right)+3\ln \left(\dfrac{1}{\cos \left(5\right)}\right)-3\ln \left(-\dfrac{1}{\cos \left(3\right)}\right)[/tex]

value:

[tex]\sf \hookrightarrow 1086950.36760[/tex]

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Answers

Step-by-step explanation:

Graph to move right

As we can see mad is same (10)

Hence it doesn't affect the mad what we do with base axis

So if we turn mean to right it gets bigger.

Option A is correct

Does the budgeted amount cover the actual amount for expenses,savings,and emergencies? A. No it’s short $207.00 B.No it’s short $227.00 C.Yes,there’s a surplus of $207.00 D yes there’s is a surplus of 227.00

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The budget was $207 short, hence option A is correct.

A similar problem, in which an amount is studied, is given at https://brainly.com/question/25537936

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Answers

Answer:

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Step-by-step explanation:

Attached below is a picture with each fraction illustrated.

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Answers

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Diagonal : 10.2

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Shorter side: 4

Step-by-step explanation:

Answer:

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Step-by-step explanation:

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Answers

Answer:

9.) 37.55102% or 37.6 depending on how deep into the question you are required to go.

10.) not sure. someone else will have to help you

Step-by-step explanation:

A medium pizza cheese pizza a papa johns costs $6.80 plus $0.90 for each topping. The cost of a medium cheese pizza at pizza hut is $7.30 plus $o.65 for each topping. How many toppings need to be added to a large cheese pizza from papa johns and pizza hut in order for the pizzas to cost the same, not including tax?

Answers

Answer:

I hope I answered your question!

Step-by-step explanation:

6.80+0.90x=7.30+0.65x (multiply by 100 on both sides)

680+90x=730+65x

50=25x

x=2

Therefore you need to add 2 toppings to each pizza to make them cost the same.

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Answers

Answer:

Step-by-step explanation:

(a)

Data Set 1

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On comparing the Mean Aboslute Deviation values of the two games, it is seen that that the Leaping Lizards game has a higher MAD than the Jumping Jackrabbits game, which means the scores deviate more in the Leaping Lizards game, whereas in the other game it is consistent

A spinner is divided into five colored sections that are not of equal size: red,
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Color Frequency
7
16
Red
Blue
Green
Yellow
Purple
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Answers

Answer:

15.09%

Step-by-step explanation:

Sum of frequencies = 53

Probability = 8

53

=0.1509

Expressing as a percentage

>> 15.09%

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Answer:

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partB option A is the correct answer

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Answers

The minimum of x is 15, and the most is infinity, which is the same thing as [15, ∞).

We have given inequality is

–3(6 – 2x) ≥ 4x + 12

What is the meaning of inequality?

The statement of an order relationship greater than, greater than or equal to, less than, or less than or equal to between two numbers or algebraic expressions.

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This means that the minimum of x is 15, and the most is infinity, which is the same thing as [15, ∞).

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Answers

Answer:

500

Remember the rounding rule for the ones place...

0 to 4? Round down or keep the same.

5 to 9? Round up.

__________________________________________________________

Questions?

Ask in comments.

It would stay 500! Hope this helped

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Answers

Answer:

let's say the Missing number is called 'y'.

product means multiplication so

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Hope this at least helps.

Step-by-step explanation:

y x 5 =8

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Answers

Answer:

D) AD ≅ AE

Step-by-step explanation:

According to given details the triangles ADB and BCE are congruent as per HL congruence

Lets verify the choices

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Step-by-step explanation:

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888 + 88 + 8 + 8 + 8 = 1000

have a nice day!

888 + 88 + 8 + 8 + 8 = 1000

hope this helps....

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Answers

Answer:

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Step-by-step explanation:

Answer:

15,600

Step-by-step explanation:

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plan c. is the answer byebye

1. X=7 and y=3

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Answers

Answer:

A. 10

B. 17

C. 29

D. 80

E. 300

Step-by-step explanation:

A.

x + y

7 + 3

10

B.

x + x + y

7 + 7 + 3

14 + 3

17

C.

2x + 5y

2(7) + 5(3)

14 + 15

29

D.

8(x + y)
8(7 + 3)

8(10)

80

E.

15(2x + y + y)

15(2(7) + 3 + 3)

15(14 + 3 + 3)

15(17 + 3)

15(20)

300

Question
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Answers

Answer:

H = 5.0

Step-by-step explanation:

Using the formula

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Solving for h

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Answer:

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Step-by-step explanation:

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Susan took out a personal loan for $3,500 at an interest rate of 13% compounded monthly. she made arrangements to pay the loan off in 3 years. what will her monthly payment be? a. $99.34 b. $105.32 c. $117.93 d. $156.60

Answers

Susan's monthly payment will be $117.93.

We have Susan take out a personal loan for $3,500 at an interest rate of 13% compounded monthly.

P=3500

r=30%

t=3

What is the amortization formula?

[tex]A = \frac{P(\frac{r}{12} )}{(1 -(1 +r/12)^{(-12t)}}[/tex]

Where A is the payment,

P= principal,

r =the annual interest rate

t is the number of years.

use the given value in the formula we get

[tex]A = \frac{3500(0.13/12)}{(1 -(1 +0.13/12)^{-36} }[/tex]

A=117.9288

A= 117.93

Susan's monthly payment will be $117.93.

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Answer: C $117.93

Step-by-step explanation:

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Answer: So Remember that the range is how far the graph goes from down to up. Look at the furthest point down on the graph or the bottom of the graph. The y-value at this point is y = 1 y=1 y=1.

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Answer:

7 hrs 42 mins

SAnswer: 7 hrs 42 mins

Step-by-step explanation:

San Antonio to Houston 3 1/3

Houston to Dallas 4 1/12

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7 5/12= 7.4166667

= 7hrs and 42 minstep-by-step explanation:

3x^2+7x+14=-7x to the nearest tenth

Answers

Answer:

hope this helps

Step-by-step explanation:

3x^2+ 14 x+14= 0

-3.22 , -1.45

Condos in Padre Island increase in value by 7% each year. If the Garza's family's condo is now worth
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Answers

257727 I used this formula y=150000(1+0.07)^8

A rectangle has the sum of length and breadth 20 cm. find the perimeter.​

Answers

Answer:

40 cm.

Step-by-step explanation:

[tex]Because=(long+width)\times 2=perimeter[/tex]

[tex]50:the\ perimeter=20\times 2=40\ cm[/tex]

I hope this helps you

:)

Which situation below has the highest constant of
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A) 16 ounces of jasmine rice costs $4.80.
B) 32 ounces of white rice costs $6.40
C) 12 ounces of white rice costs $1.20.
D) 9 ounces of brown rice costs $1.80

Answers

Answer:

A) k = 0.3

Step-by-step explanation:

Find the constant of proportionality k in each case and compare

y = kx

A)

16k = 4.80 ⇒ k = 4.80/16 ⇒ k = 0.3

B)

32k = 6.40 ⇒ k = 6.40/32 ⇒ k = 0.2

C)

12k = 1.20 ⇒ k = 1.20/12 ⇒ k = 0.1

D)

9k = 1.80 ⇒ k = 1.80/9 ⇒ k = 0.2

As we see k = 0.3 is the highest one

We need unit rates for all

#A

4.80/160.3

#B

6.4/320.2

#C

1.2/120.1

#D

1.8/90.2

A has highest

Which function in vertex form is equivalent to f(x) = x2 8 – 16x? f(x) = (x – 8)2 – 56 f(x) = (x – 4)2 0 f(x) = (x 8)2 – 72 f(x) = (x 4)2 – 32

Answers

The function in vertex form that is equivalent to the equation f(x) = x^2 + 8 - 16x from the given options is given by: Option A: f(x) = (x-8)^2 -56

What is vertex form of a quadratic equation?

If a quadratic equation is written in the form

[tex]y=a(x-h)^2 + k[/tex]

then it is called to be in vertex form. It is called so because when you plot this equation's graph, you will see vertex point(peak point) is on (h,k)

Since it is specified that the options are in vertex form, so let we convert the given equation to vertex form.

The specified equation is: [tex]f(x) = x^2 + 8 - 16x[/tex]

Converting to vertex form:

[tex]f(x) = x^2 + 8 - 16x\\\\f(x) = x^2 -16x + 8\\[/tex]

(let we make terms such that it becomes (a+b)^2 's formula).

[tex]f(x) = x^2 -16x + 8\\f(x) = x^2 -2\times8x + (-8)^2 -(-8)^2 + 8\\f(x) = (x-8)^2 -64 + 8\\f(x) = (x-8)^2 -56[/tex]

Thus, the function in vertex form that is equivalent to the equation f(x) = x^2 + 8 - 16x from the given options is given by: Option A: f(x) = (x-8)^2 -56

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Answer:

A

Step-by-step explanation:

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3 (4+13)

Step-by-step explanation:

The GCF of 12 and 39 in 3 so then you divide 12 and 39 by 3.

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In macroeconomics, a _______________ describes the common way in which market values are measured in an economy. Which type of enterprise has unlimited liability?A. corporationB. public-limited companyC. partnershipD. private-limited company Which definition best describes qualitative research?. Vectors r, s, t, u, and v are shown on the graph.Which of the following vectors is equivalent to 2/3v (Image shown) Brainstorm a list of traits for which there might be genetic variation in this clutch of owls. What is the value of x?Enter your answer in the box.x = What is the rule for 1 10 28 64 136 Write the first six terms of the numerical pattern that starts at zero and follows the rule add three Which of the following flood mitigation techniques might a city use? Select the two correct answers.A. Restricting new homes from being built on the floodplain B. Building homes on the perimeter of the floodplain C. Building homes in the middle of the floodplain D. Relocating existing homes off of the floodplain E. Relocating animals off of the floodplain Which two types of insurance are optional?A. Auto and health insuranceB. Auto and homeowners insuranceO C. Life and auto insuranceD. Health and life insurance y = 3x + 1, -x + 2y = -3 ____________ works of art are created on frosted parchment-like material or surface, which can be made from young calf skin that is stretched out.. He arrived late.His reason being that he had an appointment elsewhee New Guniea and Australia have many similar animals becuase experts think that theywere one land mass during the last ice age.TrueFalse A laborer working on an oil rig was called ____________. a. a prospector b. a roughneck c. a wildcatter d. a refiner What is the third step of the hydrogen fusion process? e v energy 2 energy energy 6 2 2 energy 2v In the Giver what does Jonas want control over? which features used to identify igneous sedimentary and metamorphic rock samples Which layer of the earth is made of hot solid iron and nickel?. use Mental Math to determine if the circumference of a circle with a radius of 5in will be greater than or less than 30 in write an argument that can be used to justify your solution