1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
7nadin3 [17]
2 years ago
15

Can someone check whether its correct or no? this is supposed to be the steps in integration by parts​

Mathematics
1 answer:
Gwar [14]2 years ago
5 0

Answer:

\displaystyle - \int \dfrac{\sin(2x)}{e^{2x}}\: \text{d}x=\dfrac{\sin(2x)}{4e^{2x}}+\dfrac{\cos(2x)}{4e^{2x}}+\text{C}

Step-by-step explanation:

\boxed{\begin{minipage}{5 cm}\underline{Integration by parts} \\\\$\displaystyle \int u \dfrac{\text{d}v}{\text{d}x}\:\text{d}x=uv-\int v\: \dfrac{\text{d}u}{\text{d}x}\:\text{d}x$ \\ \end{minipage}}

Given integral:

\displaystyle -\int \dfrac{\sin(2x)}{e^{2x}}\:\text{d}x

\textsf{Rewrite }\dfrac{1}{e^{2x}} \textsf{ as }e^{-2x} \textsf{ and bring the negative inside the integral}:

\implies \displaystyle \int -e^{-2x}\sin(2x)\:\text{d}x

Using <u>integration by parts</u>:

\textsf{Let }\:u=\sin (2x) \implies \dfrac{\text{d}u}{\text{d}x}=2 \cos (2x)

\textsf{Let }\:\dfrac{\text{d}v}{\text{d}x}=-e^{-2x} \implies v=\dfrac{1}{2}e^{-2x}

Therefore:

\begin{aligned}\implies \displaystyle -\int e^{-2x}\sin(2x)\:\text{d}x & =\dfrac{1}{2}e^{-2x}\sin (2x)- \int \dfrac{1}{2}e^{-2x} \cdot 2 \cos (2x)\:\text{d}x\\\\& =\dfrac{1}{2}e^{-2x}\sin (2x)- \int e^{-2x} \cos (2x)\:\text{d}x\end{aligned}

\displaystyle \textsf{For }\:-\int e^{-2x} \cos (2x)\:\text{d}x \quad \textsf{integrate by parts}:

\textsf{Let }\:u=\cos(2x) \implies \dfrac{\text{d}u}{\text{d}x}=-2 \sin(2x)

\textsf{Let }\:\dfrac{\text{d}v}{\text{d}x}=-e^{-2x} \implies v=\dfrac{1}{2}e^{-2x}

\begin{aligned}\implies \displaystyle -\int e^{-2x}\cos(2x)\:\text{d}x & =\dfrac{1}{2}e^{-2x}\cos(2x)- \int \dfrac{1}{2}e^{-2x} \cdot -2 \sin(2x)\:\text{d}x\\\\& =\dfrac{1}{2}e^{-2x}\cos(2x)+ \int e^{-2x} \sin(2x)\:\text{d}x\end{aligned}

Therefore:

\implies \displaystyle -\int e^{-2x}\sin(2x)\:\text{d}x =\dfrac{1}{2}e^{-2x}\sin (2x) +\dfrac{1}{2}e^{-2x}\cos(2x)+ \int e^{-2x} \sin(2x)\:\text{d}x

\textsf{Subtract }\: \displaystyle \int e^{-2x}\sin(2x)\:\text{d}x \quad \textsf{from both sides and add the constant C}:

\implies \displaystyle -2\int e^{-2x}\sin(2x)\:\text{d}x =\dfrac{1}{2}e^{-2x}\sin (2x) +\dfrac{1}{2}e^{-2x}\cos(2x)+\text{C}

Divide both sides by 2:

\implies \displaystyle -\int e^{-2x}\sin(2x)\:\text{d}x =\dfrac{1}{4}e^{-2x}\sin (2x) +\dfrac{1}{4}e^{-2x}\cos(2x)+\text{C}

Rewrite in the same format as the given integral:

\displaystyle \implies - \int \dfrac{\sin(2x)}{e^{2x}}\: \text{d}x=\dfrac{\sin(2x)}{4e^{2x}}+\dfrac{\cos(2x)}{4e^{2x}}+\text{C}

You might be interested in
Miguel is a cartoonist. Miguel drew 2 cartoon cells in 3 hours. If he keeps drawing at the same rate, about how many cells will
Sonja [21]

In three hours Miguel drew 2 cartoon cells . And since he keeps drawing at the same rate, therefore,

in one hour Miguel drew (2/3) cartoon cells .

Hence in 9 hours, Miguel drew (2/3)* 9 = 18/3=6 cartoon cells .

So the answer of the given question is 6 cartoon cells .

4 0
3 years ago
Read 2 more answers
Given the table below, determine if the data represents a linear or an exponential function and find a possible formula for the
strojnjashka [21]
The answer is C) y=14(0.9)ˣ and it's an exponential function

PROOF 

Give to x the respective values of x & calculate y, using y = 14(0.9)ˣ

x         |       y
---------|---------
0         | 14(0.9)⁰ = 14
1         | 14(0.9)¹ = 12.6
2         | 14(0.9)² = 11.34
3         | 14(0.9)³ = 10.206
4         | 14(09)⁴ =  9.1845
4 0
3 years ago
In a sample of 56 bags of fertilizer, the average weight was found to be 17.2lb with a standard deviation of 0.7. Give a point e
Natali5045456 [20]

Answer:

???????????????????????

5 0
2 years ago
Inductive reasoning involves applying a general rule to a specific situation.
Elenna [48]
I believe the correct answer is true. Inductive reasoning involves applying a general rule to a specific situation. It <span>is a logical process in which multiple premises, all believed true or found true most of the time, are combined to obtain a specific conclusion. Hope this answers the question.</span>
5 0
3 years ago
Indicate True of False for 3g &gt; 15 Substitute 5 for g.
olga55 [171]

Answer:

False

Step-by-step explanation:

(3)(5)>15

15>15

False

7 0
3 years ago
Other questions:
  • Please help me I’m desperate for help.........due tomorrow multiple choice
    8·1 answer
  • Using the distributive property to rewrite the expression as a multiple of a sum of two numbers with no common factor, how would
    13·1 answer
  • Use the unit circle to find the inverse function value in degrees.<br> tan^-1 sqrt 3
    10·1 answer
  • Is this a true proportion? 15. 22:88::10: 40
    10·1 answer
  • 2000x100 guesss yall
    8·2 answers
  • What is 5^3 x 5^11 <br> im so lost
    15·2 answers
  • 7 times what equals 17?
    11·2 answers
  • Say you want to know if boys are more likely than girls to "wait for their missionary". You take a random sample of 57 males and
    15·1 answer
  • Solve for x.
    15·1 answer
  • PLEASE HELP! DUE TODAY! TY!
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!