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
Genrish500 [490]
3 years ago
9

exFormula1" title=" \int\limits {3 ^{ \frac{x}{2} } } \, dx " alt=" \int\limits {3 ^{ \frac{x}{2} } } \, dx " align="absmiddle" class="latex-formula">=
Mathematics
2 answers:
Kryger [21]3 years ago
7 0
\int 3^{\tfrac{x}{2}}\, dx=(*)\\
t=\dfrac{x}{2}\\
dt=\dfrac{x}{2}\, dx\\
dx=2\, dt\\
(*)=\int 3^t\cdot2\, dt=\\
2\int 3^t \, dt=\\
2\cdot\dfrac{3^t}{\ln 3}+C=\\
\boxed{\dfrac{2\cdot3^{\tfrac{x}{2}}}{\ln 3}+C}

jekas [21]3 years ago
7 0
\int { { 3 }^{ \frac { x }{ 2 }  } } dx\\ \\ =\int { { \left( { 3 }^{ x } \right)  }^{ \frac { 1 }{ 2 }  } } dx

However:

u={ 3 }^{ x }\\ \\ \therefore \quad \frac { du }{ dx } ={ 3 }^{ x }\cdot \ln { 3 } \\ \\ \therefore \quad du={ 3 }^{ x }\cdot \ln { 3 } dx\\ \\ \therefore \quad dx=\frac { 1 }{ { 3 }^{ x }\cdot \ln { 3 }  } du=\frac { 1 }{ u\cdot \ln { 3 }  } du

So let's use:

\int { { u }^{ \frac { 1 }{ 2 }  } } \cdot \frac { 1 }{ u\cdot \ln { 3 }  } du\\ \\ =\int { \frac { 1 }{ \ln { 3 }  }  } \cdot { u }^{ -\frac { 1 }{ 2 }  }du

But you need to know that:

\int { k{ u }^{ n } } du\\ \\ =\frac { k{ u }^{ n+1 } }{ n+1 } +C

Therefore:

\int { \frac { 1 }{ \ln { 3 }  }  } \cdot { u }^{ -\frac { 1 }{ 2 }  }du\\ \\ =\frac { \frac { 1 }{ \ln { 3 }  } \cdot { u }^{ \frac { 1 }{ 2 }  } }{ \frac { 1 }{ 2 }  } +C

\\ \\ =\frac { 1 }{ \ln { 3 }  } \cdot { u }^{ \frac { 1 }{ 2 }  }\cdot 2+C\\ \\ =\frac { 1 }{ \ln { 3 }  } \cdot { 3 }^{ \frac { x }{ 2 }  }\cdot 2+C\\ \\ =\frac { 2\cdot { 3 }^{ \frac { x }{ 2 }  } }{ \ln { 3 }  } +C
You might be interested in
Standard Form<br> -19 - 24y+ 8y^7 help help wjeko
Iteru [2.4K]

8y7−24y−19 This is what it is in standard form

7 0
3 years ago
What is the equation of the line that passes through the point (-5,2 ) and has the slope of -2/5
valina [46]

Answer:

y = -2/5x + 2

Step-by-step explanation:

8 0
2 years ago
Simplify 2x+x+x+2y x 3y x y
olasank [31]

Answer:

3x+6y^3

Step-by-step explanation:

-First you add all the Xs

2x+x+x

-Then you multiply all the Ys

2y * 3y * y

* the Y has a 1 as an exponent by multiplying the Ys the exponents are added making the Y a Y^3

*the numbers are multiplied making it a 6

Together it forms 6y^3

-Ys and Xs cannot be added because they are different values so your answer would be 3x+ 6y^3

3 0
3 years ago
Read 2 more answers
X · 1 + = ___.<br><br> 1<br> x<br> 1 + x<br> 2x
Aliun [14]

Answer:x · 1 + = ___.

1

x

1 + x

2x

Step-by-step explanation:

7 0
2 years ago
I need help on this math promble
yulyashka [42]
The answer i think is 14
7 0
2 years ago
Other questions:
  • please calculate the volume of the cylinder below. use 3.14 as approximation for Pi. (MIDDLE SCHOOL MATH)​
    14·1 answer
  • Vidal bought 2 pizzas he cut each into 8 slices. he and his friend ate 10 slices. write an expression to match the word
    9·2 answers
  • I need to write y=-2/-3+490 is function notation
    9·1 answer
  • When the following fraction is reduced, what will be the exponent on the m? 27mn^3/56m^6n
    13·2 answers
  • How long does it take for light from a star that is 8 light years away to reach Earth?
    6·1 answer
  • Some one please help me with these questions I’m struggling :(
    13·2 answers
  • On Tuesday the temperature is 1 ºC. By Wednesday it has dropped to –2 ºC. The temperature drops by twice as much from Wednesday
    8·1 answer
  • A probability experiment will consist of selecting one of two stacks of cards, and then choosing a card at random from the stack
    10·1 answer
  • Which could be the missing data item for the given set of data if the median of the complete data set is 15?
    15·1 answer
  • Helppp meee pleasssseee!!!
    15·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!