please use an equation step by step and please show the numbers do not use words

[tex]\bold{\huge{\underline{ Solution }}}[/tex]

- I have used different colours for indicating different process

- I used brown colour to indicate steps

- I used green colour for explaining solution

- Pink colour for known properties of integration

- Purple colour for the middle steps that are optional.

[Note :- For solving such questions it is mandatory that you should know all the functions and it's derivatives ]

[tex]\sf{=}{\sf{\dfrac{ x^{2}}{2}}}{\sf{ + 7x + 67x .In| x - 4 | - 30.In| x - 3 | + c }}[/tex]

Answer:

[tex]\displaystyle \large{\frac{x^2}{2}+7x+67\ln |x-4|-30\ln |x-3| + C}[/tex]

Step-by-step explanation:

To find an integration of this function, first, you must know these integration methods and differences of them.

These methods above are common when it comes to integrating fractional functions, except there are differences when to use these methods.

Examples (of these integrands that require partial fraction technique)

[tex]\displaystyle \large{\int \frac{3}{x^2-5x+6} \ dx}\\\displaystyle \large{\int \frac{x+2}{x^2-4} \ dx}[/tex]

Examples (of these integrands that require long division)

[tex]\displaystyle \large{\int \frac{x^2+5x+6}{x+2} \ dx}\\\displaystyle \large{\int \frac{x^4+3}{x} \ dx }[/tex]

Therefore, from the given integral, the integrand requires long division method. (See attachment for long division.)

After long division, we should get:

[tex]\displaystyle \large{\int x+7+\frac{37x-81}{x^2-7x+12} \ dx}[/tex]

Recall important properties of integral such as:

[tex]\displaystyle \large{\int [f(x) \pm g(x)] \ dx = \int f(x) \ dx \pm \int g(x) \ dx}[/tex]

Hence:

[tex]\displaystyle \large{\int x \ dx + \int 7 \ dx + \int \frac{37x-81}{x^2-7x+12} \ dx}[/tex]

Recall important integration formula for polynomial and constant:

[tex]\displaystyle \large{\int x^n \ dx = \frac{x^{n+1}}{n+1} + C}\\\displaystyle \large{\int x \ dx = \frac{x^2}{2} + C}\\\displaystyle \large{\int k \ dx = kx + C \ \ \tt{(k \ \ is \ \ a \ \ constant.)}[/tex]

Therefore:

[tex]\displaystyle \large{\frac{x^2}{2}+7x+\boxed{\int \frac{37x-81}{x^2-7x+12} \ dx}}[/tex]

From the boxed integral above, we cannot evaluate it by default. From what I said, if the degree of numerator is less than the degree of denominator, we’ll use partial fraction technique.

Therefore, factor the denominator:

[tex]\displaystyle \large{\frac{37x-81}{(x-4)(x-3)}}[/tex]

Set to A/x-4 and B/x-3

[tex]\displaystyle \large{\frac{37x-81}{(x-4)(x-3)} = \frac{A}{x-4} + \frac{B}{x-3}}[/tex]

Multiply both sides by (x-4)(x-3).

[tex]\displaystyle \large{\frac{37x-81}{(x-4)(x-3)} \cdot (x-4)(x-3)= \frac{A}{x-4} \cdot (x-4)(x-3) + \frac{B}{x-3} \cdot (x-4)(x-3)}\\\displaystyle \large{37x-81= A(x-3) + B(x-4)}\\\displaystyle \large{37x-81= Ax-3A+Bx-4B}\\\displaystyle \large{37x-81= (A+B)x-(3A+4B)}[/tex]

Then compare the coefficients:

[tex]\displaystyle \large{\left \{ {{A+B=37} \atop {3A+4B=81}} \right}[/tex]

Solve the simultaneous equation for A and B.

[tex]\displaystyle \large{\left \{ {{A=37-B} \atop {3A+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {3(37-B)+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {111-3B+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {111+B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {B=81-111}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {B=-30}} \right}\\\displaystyle \large{\left \{ {{A=67} \atop {B=-30}} \right}\\[/tex]

Therefore, A = 67 and B = -30. Substitute the values in:

[tex]\displaystyle \large{\int \frac{67}{x-4}-\frac{30}{x-3} \ dx}\\\displaystyle \large{\int \frac{67}{x-4} \ dx -\int \frac{30}{x-3} \ dx}[/tex]

Recall the integration formula for above:

[tex]\displaystyle \large{\int \frac{k}{x-a} \ dx = \int k \cdot \frac{1}{x-a} \ dx = k\ln |x-a| + C}[/tex]

Therefore:

[tex]\displaystyle \large{\int \frac{37x-81}{x^2-7x+12} \ dx = \int \frac{67}{x-4}-\frac{30}{x-3} \ dx = 67\ln |x-4| - 30\ln |x-3|}[/tex]

Back from this:

[tex]\displaystyle \large{\frac{x^2}{2}+7x+\boxed{\int \frac{37x-81}{x^2-7x+12} \ dx}}[/tex]

Substitute in:

[tex]\displaystyle \large{\frac{x^2}{2}+7x+67\ln |x-4|-30\ln |x-3| + C}[/tex]

Therefore the solution is:

[tex]\displaystyle \large \boxed{\frac{x^2}{2}+7x+67\ln |x-4|-30\ln |x-3| + C}[/tex]

Answer:

16+ 2x

Step-by-step explanation:

2(10) +2(x-4)

20 + 2x -8

16+ 2x

Answer:

As it says approximate, the nearest value to 46.6° is 45.6°

Therefore, 45.6° is the approximate measure of angle A.

using SOH CAH TOA:

[tex]\sf cos(A) = \dfrac{b^2+c^2-a^2}{2bc}[/tex]

[tex]\hookrightarrow \sf cos(A) = \dfrac{4^2+8^2-6^2}{2(8)(4)}[/tex]

[tex]\hookrightarrow \sf cos(A) = 0.6875[/tex]

[tex]\hookrightarrow \sf A = cos^{-1}(0.6875)[/tex]

[tex]\hookrightarrow \sf A = 46.5675^{\circ \:}[/tex]

[tex]\hookrightarrow \sf A = 46.6^{\circ \:}[/tex]

Answer:

[tex]x = \frac{4}{3}- 1[/tex]

x=4/3 , -1

Answer:

3/5 miles

Step-by-step explanation:

Average Distance = Total Distances / Total Number of Runners

Average Distance = [0.3+0.4+0.5+0.6+(4*0.7)+0.8] / 9 = 3/5

Answer:

for the values 45, 81

The factors of 45 are: 1, 3, 5, 9, 15, 45

The factors of 81 are: 1, 3, 9, 27, 81

Then the greatest common factor is 9.

Answer:

Step-by-step explanation:

Factor of 45: → 1, 3, 5, 9, 15, 45

Factor of 81: →  1, 3, 9, 27,81

Therefore, GCF of 45 and 81 is 9

~

Answer:

the answer send me a 100dollar gift card

Answer:

10 = 3 hours

one hour = 3.33 dollars

Step-by-step explanation:

Answer:

( 7/18 )

Step-by-step explanation:

7           4            7 x 4            28 ÷ 4                 7

/    x     /      =        /         =          /              =      /

8          9             8 x 9           72 ÷ 4                18

I hope this helps!

Answer:

[tex]\tt x=21[/tex]

Step-by-step explanation:

[tex]\tt 45+63+4x-12=180[/tex]

[tex]\tt \cfrac{4x}{4} =\cfrac{180-96}{4}[/tex]

[tex]\tt x=21[/tex]

~

Answer:

It's asking what 3^k+2 is in terms of m. That is where the 2 comes from.

Please let me know if this helped <3

Answer:

x-intercept is 5

y-intercept is -6

Step-by-step explanation:

To find x-intercept put y=0

since 6x-5y=30 then 6x=30

x=30/6 = 5

To find y-intercept put x=0

since 6x-5y=30 then -5y=30

y=30/-5 = -6

Answer:

69-46x y57 -8r8o 98-225+

Answer:

[tex]46\frac{1}{8}[/tex]

Step-by-step explanation:

Formulate

[tex]1845\div40[/tex]

Calculate

[tex]\frac{1845}{40}[/tex]

Cross out the common factor

[tex]\frac{369}{8}[/tex]

Write [tex]\frac{369}{8}[/tex] as a mixed fraction

[tex]46\frac{1}{8}[/tex]

I hope this helps you

:)

Answer:

It grew by 500% meaning it grew on average 83.333% per week

Step-by-step explanation:

To find the percent growth you divide the end weight by the starting weight:

[tex]7.5/1.5=5[/tex]

However, that is the percentage in decimal form. To convert that number to a percentage move the decimal place to the right 2 places. This results in 500%

To get the average weekly growth weight, simply divide the 500% by the amount of weeks (6). This then gives the 83.333% per week.

Note: I didn't know which answer you were looking for. That is why there are 2.

Answer:

[tex] - 4 \: \: \: \: \: 2 \ \\ - 3 \: \: \: 0[/tex]

Step-by-step explanation:

he has 60%

so he needs 40%

40% of 120

of is ×

40% × 120

40% as a decimal is

put answer in comments

Answer:

The answer should be 24

Step-by-step explanation:

All you need to do is replace the variables (a, b, and c) with the numbers given.

16 +64/8 would equal to 24

Answer:

58.45

Step-by-step explanation:

Use BIDMAS

14.5 X 3.1 = 44.95

44.95 + 13.5 = 58.45

Answer:

58.45

Step-by-step explanation:

Answer:

do you already have it answer.lol.

Answer:

x = 7

Step-by-step explanation:

8 = x + 1

prove that the value of x is 7

proven

Answer:

Its C: 12 units × 8 units are the dimensions of the rectangle

Step-by-step explanation:

Answer:

- Parallelogram has two pairs of parallel sides.

- Square has four right angles, two pairs of parallel sides, and all sides congruent.

- Trapezoid has only one pair of parallel sides.

Step-by-step explanation:

Answer:

[tex]x=20[/tex]

Based on the given conditions, formulate:

[tex]62\times[/tex] [tex]x=1240[/tex]

Solve the equation

[tex]62x=1240[/tex]

Divide both sides of the equation by the coefficient of variable

[tex]x=\frac{1240}{62}[/tex]

Cross out the common factor

[tex]x=20[/tex]

I hope this helps you

:)

Answer:

6/8 red paint

10/8 blue paint

4th grade savvas realize

Answer:

100 + 50p = t or t = 100 + 50p

Step-by-step explanation:

We know that they charge 100 initially and then an addition 50 for each person.

We add 100 to 50 but we do not know the amount of people, which leaves it unknown (p)

The total cost (t) is also unknown which stays unknown until we calculate the amount of people.

Answer:

mDBC = 341 °

Step-by-step explanation:

Arc DBC is almost the whole circle, if you were to exclude the degree measure of minor arc DC; which is, in other words m∠ DPC

Now by Vertical Angles Theorem:

m ∠ DPC = m∠ APB, ⇒ provided BD, and AC are diameters

m∠ DPC = 19°

This would mean ⇒

mDBC = 360 - m∠ DPC,

mDBC = 360 - 19,

mDBC = 341 degrees ( ° )

Answer:

it's a it's the first one

Step-by-step explanation:

because if the trick would use less oil then that would make it look like the truck is less expensive but it's not

step 1

[tex]\begin{matrix}\:\:\:\:\emptyspace0.\:\:\:\:\:\:\:\:\:\:\:\\ 8\overline{|\smallspace3.0}\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace0}\:\:\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\emptyspace3\emptyspace0\:\:\:\:\:\:\:\:\:\:\end{matrix}[/tex]

step 2

[tex]\begin{matrix}\:\:\:\:\emptyspace0.\emptyspace3\:\:\:\:\:\:\:\:\:\\ 8\overline{|\smallspace3.0\cdot \:0}\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace0}\:\:\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\emptyspace3\emptyspace0\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace2\emptyspace4}\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\emptyspace6\emptyspace0\:\:\:\:\:\:\:\:\end{matrix}[/tex]

step 3

[tex]\begin{matrix}\:\:\:\:\emptyspace.\emptyspace3\emptyspace7\:\:\:\:\:\:\:\\ 8\overline{|\smallspace 3 0\ \ \:}\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace0}\:\:\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\emptyspace3\emptyspace0\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace2\emptyspace4}\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\emptyspace6\emptyspace0\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\underline{\emptyspace5\emptyspace6}\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\:\:\emptyspace4\emptyspace0\:\:\:\:\:\:\end{matrix}[/tex]

step 4

[tex]\begin{matrix}\:\:\:\:\epe0.\ece3\empt7\e5\:\:\:\:\:\\ 8\overline{|\sm30 \ \ \ \ }\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace0}\:\:\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\em3\e0\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace2\emptyspace4}\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\eme6\ece0\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\underline{\empte5\ee6}\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\:\:\ee4\ece0\:\:\:\:\:\:\\ \:\:\:\:\:\:\:\:\underline{\ee4\epace0}\:\:\:\:\:\:\\ \:\:\:\:\:\:\:\:\:\:\e0\:\:\:\:\:\:\end{matrix}[/tex]

Answer:

3.75

Step-by-step explanation:

[tex]\mathrm{Write\:the\:problem\:in\:long\:division\:format}[/tex]

[tex]\begin{matrix}8\overline{|\smallspace3.0}\:\:\:\:\:\:\:\:\:\:\:\:\end{matrix}[/tex]

[tex]\mathrm{Divide\;30\;by\;8\;to\;get\;3}[/tex]

[tex]\begin{matrix}\:\:\:\:\:\:\emptyspace3.\:\:\:\:\:\:\:\:\:\:\:\\ 8\overline{|\smallspace30.0}\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace2\emptyspace4}\:\:\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\emptyspace6\emptyspace0\:\:\:\:\:\:\:\:\:\:\end{matrix}[/tex]

[tex]\mathrm{Divide\;60\;by\;8\;to\;get\;7}[/tex]

[tex]\begin{matrix}\:\:\:\:\:\:\emptyspace3.\emptyspace7\:\:\:\:\:\:\:\:\:\\ 8\overline{|\smallspace30.0\cdot \:0}\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace2\emptyspace4}\:\:\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\emptyspace6\emptyspace0\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\underline{\emptyspace5\emptyspace6}\:\:\:\:\:\:\:\:\:\:\\ \:\:\:\:\:\:\:\:\emptyspace4\emptyspace0\:\:\:\:\:\:\:\:\end{matrix}[/tex]

[tex]\mathrm{Divide\;40\;by\;8\;to\;get\;5}[/tex]

[tex]\begin{matrix}\:\:\:\:\:\:\emptyspace3.\emptyspace7\emptyspace5\:\:\:\:\:\:\:\\ 8\overline{|\smallspace30.0\cdot \:0}\:\:\:\:\:\:\:\\ \:\:\:\:\underline{\emptyspace2\emptyspace4}\:\:\:\:\:\:\:\:\:\:\:\:\\ \emptyspace6\emptyspace0\:\:\:\:\:\:\:\:\:\:\\ \underline{\emptyspace5\emptyspace6}\:\:\:\:\:\:\:\:\:\:\\ \emptyspace4\emptyspace0\:\:\:\:\:\:\:\:\\ \underline{\emptyspace4\emptyspace0}\:\:\:\:\:\:\:\:\\ \emptyspace0\:\:\:\:\:\:\:\:\end{matrix}[/tex]

[tex]\mathrm{The\:solution\:for\:Long\:Division\:of}\:\frac{3}{0.8}\:\mathrm{is}\:3.75[/tex]

~lenvy~