Answers
[tex]\bold{\huge{\underline{ Solution }}}[/tex]
Here, we will use the concept of integration and algebric identities
Integration is the process of finding function that is a derivate of given function Three important trigonometric identities :-[tex]\sf{ sin^{2}{\theta}+ cos^{2}{\theta} = 1 }[/tex][tex]\sf{ 1 + tan^{2}{\theta} = sec^{2}{\theta} }[/tex][tex]\sf{ 1 + cot^{2}{\theta} = cosec^{2}{\theta} }[/tex]Let's Begin :-We have,
[tex]\bold{\displaystyle\int}{\bold{\dfrac{ cos^{2}x}{1 + sinx}}}{\bold{dx}}[/tex]
By using trigonometric identity,
[tex]\sf{ sin^{2}{\theta}+ cos^{2}{\theta} = 1 }[/tex][tex]\sf{\displaystyle\int}{\sf{\dfrac{ 1 -sin^{2}x }{1 + sinx}}}{\sf{dx}}[/tex]
By using algebraic identity :-
[tex]\sf{ a^{2} - b^{2} = ( a + b) (a - b) }[/tex][tex]\sf{\displaystyle\int}{\sf{\dfrac{ (1 + sinx) (1 - sinx) }{1 + sinx}}}{\sf{dx}}[/tex]
[tex]\sf{\displaystyle\int}{\sf{( 1 - Sinx)dx}}[/tex]
[tex]\sf{ {\displaystyle\int} dx - {\displaystyle\int}sinxdx}[/tex]
We know that,
[tex]\sf{{\int\displaystyle}sin{\theta} d{\theta} = - cos {\theta} + C }[/tex][tex]\sf{ x - (-cosx) + c}[/tex]
[tex]\bold{ x + cosx + c}[/tex]
Hence, The answer is x + Cosx + c.
We are given with an integral and need to solve the integral , so let's start ;
[tex]{:\implies \quad \displaystyle \int \sf \dfrac{\cos^{2}(x)}{1+\sin (x)}dx}[/tex]
As we know that sin²(x) + cos²(x) = 1 , using this
[tex]{:\implies \quad \displaystyle \int \sf \dfrac{1-\sin^{2}(x)}{1+\sin (x)}dx}[/tex]
Can be further written as
[tex]{:\implies \quad \displaystyle \int \sf \dfrac{1^{2}-\sin^{2}(x)}{1+\sin (x)}dx}[/tex]
[tex]{:\implies \quad \displaystyle \int \sf \dfrac{\cancel{\{1+\sin (x)\}}\{1-\sin (x)\}}{\cancel{\{1+\sin (x)\}}}dx\quad \qquad \{\because a^{2}-b^{2}=(a+b)(a-b)\}}[/tex]
[tex]{:\implies \quad \displaystyle \int \sf \{1-\sin (x)\}dx}[/tex]
Now , as integrals follow distributive property , so ;
[tex]{:\implies \quad \displaystyle \int \sf 1\: dx-\int \sin (x)dx}[/tex]
Now , as antiderivative (Integration) of sin(x) is -cos(x) + C and that of dx is x + C So ;
[tex]{:\implies \quad \displaystyle \bf \therefore \underline{\underline{\int \bf \dfrac{\cos^{2}(x)}{1+\sin (x)}=x+\cos (x)+C}}}[/tex]
This is the Required answer
Related Questions
Write a mixed number to describe the length of the ribbon shown in the figure below.
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Step-by-step explanation:
what did we learn in the other question ?
there are 8 equal sub-segments of 1/8 length between neighboring whole numbers.
so, each little line is 1/8.
the marker goes first to 1, and then one little line towards 2.
therefore, that number is 1 1/8.
eight multiply by 5 over 12 equals
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Hi! I hope you're enjoying your day!
First, 8 multiplied by 5 is equal to 40.
Then, 40 over 12 is the same as 40 divided by 12.
But hey! 40 isn't divisible by 12.
So we write this as a fraction:
[tex]\bold{\displaystyle\frac{40}{12} }[/tex]
Surprise! This fraction can be reduced.
Divide both the numerator (40) and the denominator (12) by 4:[tex]\bold{\displaystyle\frac{10}{3} }[/tex]
Hence, the answer is:
[tex]\boxed{\boxed{\bold{\displaystyle\frac{10}{3} }}}[/tex] <-- the answer written as an improper fraction
[tex]\boxed{\boxed{\bold{3\displaystyle\frac{1}{3}}}}[/tex] <-- the answer written as a mixed number
Hope everything is clear.
If you have any questions, please comment.
#LearningIsFun
[tex]\rule{250}{2}[/tex]
i dont understand this
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PLEASE HURRYYYYYY!!!!!
A salesman sold $65,100 in bikes. He eared a 20% Commission, How much did he earn?
explain your answer please
Answers
Answer:
$13,020
Step-by-step explanation:
65,100 * 20% = 65,100 * 0.20 = 13,020
evaluate the expression 10^2+(3+5)^2-5
Answers
Hey there!
10^2 + (3 + 5)^2 - 5
= 10 * 10 + (3 + 5)^2 - 5
= 100 + (8)^2 - 5
= 100 + (8 * 8) - 5
= 100 + 64 - 5
= 164 - 5
= 159
Therefore, your answer is: 159
Good luck on your assignment & enjoy your day!
~Amphitrite1040:)
Answer:
10² + (3 + 5)² - 5 = 159
Step-by-step explanation:
Given expression,
→ 10² + (3 + 5)² - 5
Evaluating the expression,
→ 10² + (3 + 5)² - 5
→ 10² + 8² - 5
→ 100 + 64 - 5
→ 100 + 59
→ 159
Hence, the answer is 159.
solve the inequality |x+6-|3x+6||+|x+2|-x-4 less then or equal to 0
Answers
It looks like the inequality is
[tex]\bigg|x + 6 - |3x + 6|\bigg| + |x + 2| - x - 4 \le 0[/tex]
or equivalently,
[tex]\bigg|x + 6 - 3 |x + 2|\bigg| + |x + 2| - x - 4 \le 0[/tex]
Recall the definition of absolute value:
[tex]|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}[/tex]
By this definition, we have
[tex]|x + 2| = \begin{cases} x + 2 & \text{if } x + 2 \ge 0 \\ -(x + 2) & \text{if } x + 2 < 0 \end{cases} = \begin{cases} x + 2 & \text{if } x \ge -2 \\ -x - 2 & \text{if } x < -2 \end{cases}[/tex]
Suppose [tex]x < -2[/tex]. Then
[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |x + 6 - 3 (-x - 2)| + (-x - 2) - x - 4 \\\\ = |4x + 12| - 2x - 6[/tex]
By definition of absolute value,
[tex]|4x + 12| = 4 |x + 3| = \begin{cases} 4x + 12 & \text{if } x \ge -3 \\ -4x - 12 & \text{if } x < -3 \end{cases}[/tex]
so we further suppose that [tex]x < -3[/tex]. Then
[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |4x + 12| - 2x - 6 \\\\ = (-4x - 12) - 2x - 6 \\\\ = -6x - 18 \le 0 \\\\ \implies x \ge -3[/tex]
but this is a contradiction, so this case gives no solution.
If instead we have [tex]-3 \le x < -2[/tex], then
[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |4x + 12| - 2x - 6 \\\\ = (4x + 12) - 2x - 6 \\\\ = 2x + 6 \le 0 \\\\ \implies x \le -3[/tex]
and [tex]-3 \le x[/tex] and [tex]x \le -3[/tex] means that [tex]\boxed{x = -3}[/tex].
Now suppose [tex]x \ge -2[/tex]. Then
[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = |x + 6 - 3 (x + 2)| + (x + 2) - x - 4 \\\\ = |-2x| - 2 \\\\ = 2 |x| - 2[/tex]
If we further suppose that [tex]-2 \le x < 0[/tex], then
[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = 2 |x| - 2 \\\\ = -2x - 2 \le 0 \\\\ \implies x \ge -1[/tex]
Otherwise, if [tex]x \ge 0[/tex], then
[tex]\bigg| x + 6 - |3x + 6| \bigg| + |x + 2| - x - 4 \\\\ = 2 |x| - 2 \\\\ = 2x - 2 \le 0 \\\\ \implies x \le 1[/tex]
Taken together, this case gives the solution [tex]\boxed{-1 \le x \le 1}[/tex]
So, the overall solution to the inequality is the set
[tex]\boxed{\left\{ x \in \mathbb R : x = -3 \text{ or } -1 \le x \le -1 \right\}}[/tex]
Help please. Very easy
Which description can be written as the expression 6b+5?
the difference of six times a number and five
the sum of six times a number and five
the quotient of six times a number and five
the product of six times a number and five
Thanks.
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explanation: b is your variable which is going to by multiplied by 6, since there is no other symbol next to it. afterwards, you would add 5 to get your final answer.
raise v to the 8th power, triple the result, then subtract w from what you have
Answers
Answer:
Step-by-step explanation:
[tex]3(v^{8}) -w\\ 3v^{8}-w[/tex]
Two stores are having sales on cameras. • At Store X all cameras are on sale for 15% off the original price.
• At Store Y all cameras are on sale for off the original price. Which store would have the better sale price for a camera with an original price of $80?
Answers
Answer:
Store X, because the sale price would be $68
Step-by-step explanation:
vStore X, because the sale price would be $68
Assuming at Store Y all cameras are on sale for off the original price. The store with the better sale price is: Store y, because the sales price is $64.
Store x and yStore x
x=80×(1-0.15)
x=80×0.85
x=68
Store y
y=80×(-1/5+1)
y=80×4/5
y=64
y=64≤68
Based on the above calculation store y is the best.
Inconclusion the store that would have the better sale price is: Store y, because the sales price is $64.
Learn more about store x and y here:https://brainly.com/question/20455611
the sum of three consecutive numbers is 12 find the small number
Answers
Answer:
3
Step-by-step explanation:
Numbers include 3,4,5
smallest is 3
The circumference of a roll of party streamers is 11 inches. The circumference decreases by 3 inches after you use the streamers to decorate for a birthday party. Find the diameter of the roll after the decrease. Round your answer to the nearest hundredth.
Answers
Answer:
2.55
Step-by-step explanation:
First take 11 and subtract it by 3 to get 8.
Then divide it by 3.14 and get 2.547770701...
After that round it to 2.55
What is the circumference of a pizza that has the area of 50.3 inches?
Answers
Answer:
C = 25.1
Step-by-step explanation:
Use the formulas:
A = π r ²C = 2 π rSolving for C:
C = 2√πA = 2 · √π ·50.3 ≈ 25.14137Round to the nearest tenth:
Result = 25.1
Please helppppppppppppp
Answers
The line DB is 8cm which is made up of DE and EB which are equal to each other, thus DE and DB are both 4 cm.
Secondly, since AC and DB are perpendicular, that means angle DEC and angle DEA are ninety degrees.
Thus using the Pythagorean theorem, we can find the length of AE and EC which we can add up to find the length of AC.
[tex]AE = \sqrt{AD^2-DE^2} =\sqrt{5^2-4^2} =\sqrt{25-16} =\sqrt{9}=3\\ \\EC=\sqrt{DC^2-DE^2}=\sqrt{6^2-4^2}=\sqrt{36-16} =\sqrt{20} =2\sqrt{5} =4.47[/tex]
By adding AE and EC, we get that the length of AE and EC is 7.47 or as approximated to the nearest tenth 7.5.
Hope that helps!
NEED HELP QUICK WILL GIVE OUT 30 POINTS
Answers
4
1-
6
Step-by-step explanation:
its because i cant understand
Answer:
[tex]\Large \displaystyle \frac{7}{8}[/tex]
Step-by-step explanation:
Volume = area of cross-section × height
Mixed to improper fraction
7/4 × 1/2 = 7/8
The Jurassic Zoo charges $8 for each adult admission and $6 for each child. The total bill for the 119 people from a school trip was $754. How many adults and how many children went to the zoo?
PLEASE HELP!!! I cant figure this out!!!
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Zoe invested $16,000 in an account paying an interest rate of 3.6% compounded quarterly. Assuming no deposits or withdrawals are made, how much money, to the nearest hundred dollars, would be in the account after 20 years?
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[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$16000\\ r=rate\to 3.6\%\to \frac{3.6}{100}\dotfill &0.036\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &20 \end{cases} \\\\\\ A=16000\left(1+\frac{0.036}{4}\right)^{4\cdot 20}\implies A=16000(1.009)^{20}\implies A\approx 19100[/tex]
Find the value of 2/5 -3/5
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Which relationships describe angles 1 and 2?
Select each correct answer.
O supplementary angles
O complementary angles
O vertical angles
O adjacent angles
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Round 58.5 to the nearest whole number
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This is because we would round the decimal up to a whole number. The decimal .5 would round up, since numbers 5 and greater round up.
In the formula l2 = h2 +r2, l=25 and r=7, then the value of h is
Answers
Answer:
18
Step-by-step explanation:
I2=h2+r2
25(2)=h2+7(2)
50=h2+14
-14 -14
36=h2
÷2 ÷2
18=h
whats (x-3/4)^2 - 5 = 4 using the square root property?
please have an explanation; will give brainliest
answer: -9/4 and 15/4
Answers
Answer:
x = {-9/4, 15/4}
Step-by-step explanation:
To use the square root property to solve the quadratic, you need to get it to a form where the square root function can be usefully applied. That will mean having a square of a binomial on one side of the equal sign, and a constant on the other side of the equal sign.
We can get that from the given equation by adding 5 to both sides.
(x -3/4)^2 -5 = 4 . . . . . . given
(x -3/4)^2 = 9 . . . . . . . add 5 to get the square by itself
x -3/4 = ±3 . . . . . . . . . take the square root
x = ±3 +3/4 = {-9/4, 15/4} . . . . add 3/4 and expand the result
Complete this word
[tex] _ r _ch_i_ [\tex]
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Answer:
Arachnid is my guess.
Analyze the diagram below and complete the instructions that follow.
Find the value of x and the value of y.
Answers
Answer:
B
Step-by-step explanation:
This is an isosceles triangle. So, the sides opposite to equal angles are equal.
y = 5
To find x, use Pythagorean theorem,
x² = 5² + 5²
= 25 + 25
= 50
[tex]x =\sqrt{50}\\\\x=\sqrt{2*5*5}\\\\x=5\sqrt{2}[/tex]
Findthe y -intercept oftheparabolay = x2 + 5x + 9.
Answers
Answer:
x = 5 ± i√11 / 2
Step-by-step explanation:
At the y-intercept the value of y is equal to 0.
=> 0 = x² + 5x + 9
=> x = -5 ± √25 - 4(9) / 2
=> x = 5 ± √-11 / 2
=> x = 5 ± i√11 / 2
good estimate of the length of a standard bed pillow
Answers
Answer:
I would say 2 ft
Step-by-step explanation:
Have amazing day!
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James Marbury is the payroll accountant at All's Fair Gifts The employees of All's Fair Gifts are paid semimonthly. Lauren Stenn comes to him on April 2 and requests a pay advance of $1000, which Lauren will pay back in equal parts on the April 15 and May 15 paychecks Lauren is single with one dependent under 17 and is paid $51,680 per year Lauren contributes 3 percent of gross pay to a 401(k) and has $139 per paycheck deducted for a Section 125 plan.
Answers
Based on Lauren's income and the various deductions, the net pay for April is $1,151.44.
What is the taxable income?Taxable income:
= Semimonthly income - 401k - Section 125 plan
= (51,680 / 24 pay periods) - (0.03 x 51,680/24 periods) - 139
= 2,153.33 - 64.6 - 139
= $1,949.73
What is Lauren's April net pay?= Taxable income - federal income tax - state income tax - social security - Medicare - advance repayment
= 1,949.73 - (1.6% x 1,949.73) - (5.25% x 1,949.73) - (6.2% x 2,153.33 gross income) - (1.45% x 2,153.33) - 500
= $1,151.44
Find out more on taxable income at https://brainly.com/question/10137785.
A flower pot has a circumference of 47 inches. What is the approximate length of the radius of the flower pot? Use 3.14 for .
please answer
Answers
Answer:
7.48
Step-by-step explanation:
C=2πr
Solving for
r=C
2π=47
2·π≈7.48028
Answer:
Step-by-step explanation:
Circumference = 47 inches
[tex]2\pi r = 47\\\\2*3.14*r = 47\\\\6.28*r = 47\\\\r = \dfrac{47}{6.28}\\\\[/tex]
r = 7.48 in
explain turning point in graphs
Answers
A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum).
There are 5 subjects each school day. Each subject is 45 minutes long. Lunch and recess are 35 minutes long. How long is the school day?
Answers
Answer:
4 hours and 33 minutes
Step-by-step explanation:
Before I answer, I was confused if recess and lunch are 35 minutes in total or 35 minutes each. My answer is if they are 35 minutes combined. If that's wrong, you can just add 35 minutes to my answer.
5*45 = 225
225+35 = 260 minutes or 4 hours and 33 minutes
Please please can someone help me !!! math question !!!!
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Using the graph, it is found that:
a) The local minimum value of the function is of y = 0.
b) The values are x = -3, x = 3.
How to find the local minimums of a function is a graph?The local minimum of a function in a graph is the point at which the function changes from decreasing to increasing. In this problem, they are given by y = 0.
The values at which this happen is the values of x, that is, the horizontal axis. In this problem, they are at x = -3 and x = 3.
More can be learned about functions at https://brainly.com/question/25537936