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
Marat540 [252]
2 years ago
7

Substitution to evaluate the indefinite integral

Mathematics
1 answer:
gregori [183]2 years ago
3 0

Answer:

\displaystyle \int {e^{5x}} \, dx = \boxed{ \frac{e^{5x}}{5} + C }

General Formulas and Concepts:
<u>Calculus</u>

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:
\displaystyle (cu)' = cu'
Derivative Rule [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

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

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

Integration Methods: U-Substitution

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify given.</em>

\displaystyle \int {e^{5x}} \, dx

<u>Step 2: Integrate Pt. 1</u>

<em>Identify variables for u-substitution</em>.

  1. Set <em>u</em>:
    \displaystyle u = 5x
  2. [<em>u</em>] Differentiate [Derivative Rules and Properties]:
    \displaystyle du = 5 \ dx

<u>Step 3: Integrate Pt. 2</u>

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
    \displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\\end{aligned}
  2. [Integral] Apply Integration Method [U-Substitution]:
    \displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\& = \frac{1}{5} \int {e^u} \, du \\\end{aligned}
  3. [Integral] Apply Exponential Integration:
    \displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\& = \frac{1}{5} \int {e^u} \, du \\& = \frac{e^u}{5} + C \\\end{aligned}
  4. [<em>u</em>] Back-substitute:
    \displaystyle \begin{aligned}\int {e^{5x}} \, dx & = \frac{1}{5} \int {5e^{5x}} \, dx \\& = \frac{1}{5} \int {e^u} \, du \\& = \frac{e^u}{5} + C \\& = \boxed{ \frac{e^{5x}}{5} + C } \\\end{aligned}

∴ we have used substitution to <em>evaluate</em> the indefinite integral.

---

Learn more about integration: brainly.com/question/27746495

Learn more about Calculus: brainly.com/question/27593180

---

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

Unit: Integration

You might be interested in
Whats number two if someone can help thanks
OLEGan [10]
It would be c becuase 45/3=15
4 0
3 years ago
Henry was building a tent and had four spikes that he had to pound into the ground. The net of one of the spikes is shown. What
Misha Larkins [42]
It is exactly 4008 and thats why people love you have a nice day ^-^
3 0
3 years ago
Please answer me QICK.<br> Can the square root also be a decimal number?
Sladkaya [172]

Answer:

yes

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Help urgently please!
Nataliya [291]
1.
Slope-300.
Y int: 40,000
Y=300x+40000
2.
S: -5
Y: 250
Y=-5x+250
3.
S: 2
Y: 5
Y=2x+5
4.
S: 150
Y: 350
Y=150x+350
The last question is cut off.
5 0
3 years ago
What is 850,238x286,493?
Eddi Din [679]

Answer:

243,58,723,5,334 i'm not really good at putting ,'s in the right place.

6 0
3 years ago
Other questions:
  • Two baskets contain snacks. The basket contain 12 bags of peanuts, of which 5 are reduced sodium and 8 granola bars, of which 2
    14·1 answer
  • Use the graph of f(x) to evaluate the following:
    5·1 answer
  • How to find x and y in Triangle congruence
    11·1 answer
  • there are three source of resistance in parallel circuit. Two of them are rated at 20 ohms, the other at 10 ohms. What is the ci
    15·1 answer
  • Write the sum using summation notation, assuming the suggested pattern continues. 8 - 40 + 200 - 1000 +
    11·1 answer
  • What is the area of a parallelogram with a height of 6 feet and a base of 10.5 feet?
    13·2 answers
  • Dlane, Jim, and Ravi have a total of $62 in their wallets. has 2 times what Ravi has. Ravi has $10 more than Dlane How much do t
    14·1 answer
  • Is -45.2 an integer​
    9·1 answer
  • Solve for x Solver for x
    7·1 answer
  • Please help this makes no sense
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!