Before walking to discover the integral the e come the 2x, let united state recall a couple of facts around e2x. That is one exponential role because the base, i m sorry is 'e', is a constant (which is recognized as Euler's number) and also its exponent has actually a change in it. It deserve to be created as e^2x also. We apply the approaches of integrating exponential attributes to find the integral the e come the 2x.
You are watching: Integral of e^-2x
Let us uncover the integral that e come the 2x utilizing multiple approaches and additionally solve a few examples utilizing this.
|1.||What is the Integral of e come the 2x?|
|2.||Integral the e come the 2x by Differentiation|
|3.||Integral that e to the 2x through Substitution Method|
|4.||Integral of e to the 2x Verification|
|5.||Definite Integral the e come the 2x|
|6.||FAQs top top Integral of e come the 2x|
What is the Integral that e come the 2x?
The integral the e to the 2x is e2x/2 + C. This is mathematically created as ∫ e2x dx = e2x/2 + C. Here,e2x the is beside dx is the integrand.
Common Misconception around Integral that e to the 2x
Since ∫ ex dx = ex + C, carry out NOT think that ∫ e2x dx is e2x + C. We always have to divide the actual value of integral through the coefficient the x. Because the coefficient of x is 2, ∫ e2x dx = e2x/2 + C.
Let us prove the integral of e come the 2x to it is in e2x/2 + C using various techniques and additionally we will verify the an outcome using differentiation.
Integral the e to the 2x through Differentiation
We know that differentiation and also integration space the train station operations of every other. Also, we understand that the basic theorem the calculus is provided to discover the integral the a derivative. This organize says, ∫ f'(x) dx = f(x) + C. Well, we will first find the derivative of e2x.
By chain rule,
(e2x)' = 2e2x
Dividing both sides by 2,
(e2x)' / 2 = e2x
By consistent multiplication dominion of derivatives,
(e2x / 2)' = e2x
Taking integral top top both sides,
∫ (e2x / 2)' dx = ∫ e2x dx
Now, by the basic theorem that calculus, the integral and derivative symbols get canceled with each other on the left side and also we will certainly be left with:
e2x / 2 + C = ∫ e2x dx
Integral that e to the 2x by Substitution Method
We can discover the integral the e to the 2x utilizing the substitution technique of integration. Take into consideration the integral ∫ e2x dx. Here, we assume that 2x = u. Differentiating on both sides, we obtain 2 dx = du and it can be composed as dx = du/2. Then the over integral becomes:
∫ eu (du/2) = (1/2) ∫ eu du
We know that the integral of ex is ex + C. Utilizing this, the above integral becomes
= (1/2) (eu + C(_1))
= (1/2) eu + C(_1)/2
= (1/2) e2x + C (where C(_1)/2 = C and u = 2x)
Hence, we have actually proved the ∫ e2x dx = (1/2) e2x + C by making use of the substitution method.
Integral of e to the 2x Verification
Since the integrals and also derivatives that inverses of each other, to verify the integral of e come the x to it is in e2x / 2 + C, we need to prove that the derivative of e2x / 2 + C to it is in e2x. Let us discover the derivative.
d/dx (e2x / 2 + C)= d/dx(e2x / 2) + d/dx(C)= (1/2) (2e2x) + 0 (by the chain rule)= e2x
Hence, we have actually verified the integral of e2x.
Definite Integral the e to the 2x
A definite integral is one integral through the border (lower and also upper bounds). Us will take into consideration the definite integral the e to the 2x indigenous a to b. I.e., ∫ₐb e2x dx. To advice this, we will very first consider the fact that the integral the e2x is e2x/2 + C and also then instead of the upper bound and lower bound one after the other in order and also then subtract the results. I.e.,
∫ₐb e2x dx = (e2x/2 + C)ₐb
= (e2b/2 + C) - (e2a/2 + C)
= e2b/2 + C - e2a/2 - C
= e2b/2 - e2a/2
= (1/2) (e2b - e2a)
Thus, the integration constant doesn't beat any function while calculating the identify integral (because it acquired canceled).
Important note on Integral the e to the 2x:The integral that e to the 2x is e2x/2 + C, whereby C is the integration constant.Hence, the integral that eax, in general, is eax/a + C.Extending this further, the integral that eax+b is eax+b/a + C.
Solved examples on Integral the e to the 2x
Example 1: discover the integral of e to the 2x + 3.
Let us uncover ∫ e2x+3 dx by substitution technique of integration. For this, we assume the 2x+3 = u. Climate 2 dx = du. Indigenous this, we have, dx = du/2.
The over integral becomes:
∫ eu (du/2)
= (1/2) ∫ eu du
=(1/2) eu + C
= (1/2) e2x+3+ C (u = 2x+3 is substituted back)
Answer: ∫ e2x+3 dx = (1/2) e2x+3+ C.
Example 3: What is the integral that e come the strength of -2x from 0 come ∞.
First we will evaluate ∫ e-2x dx. Because that this, permit -2x = u ⇒ -2 dx = du ⇒ dx = -du/2. Climate the integral becomes:
∫ eu (-du/2) = (-1/2) ∫ eu du = (-1/2) eu = (-1/2) e-2x.
Now, we use the limits.
∫₀∞ e-2x dx = (-1/2) (e-2∞- e0)
= (-1/2) (0 - 1) (Because e-2∞ = 1/(e2∞) = 1/∞ = 0)
Answer: ∫₀∞ e-2x dx = 1/2.
View price >
go come slidego come slidego to slide
Have questions on straightforward mathematical concepts?
Become a problem-solving champ using logic, no rules. Find out the why behind math with our certified experts.
Book a free Trial Class
Practice inquiries on Integral of e come the 2x
Check prize >
go to slidego to slide
FAQs top top Integral that e come the 2x
What is the worth of the Integral that e come the 2x?
The integral of e^2x is e^2x/2 + C. We deserve to write this mathematically utilizing the integration symbol together ∫ e2x dx = e2x/2 + C.
How to uncover the Integral that e to the strength of 2x?
To uncover the ∫ e2x dx, assume that 2x = u. Then 2 dx = u (or) dx = du/2. Climate the worth of the integral is, (1/2) ∫ eu dx = (1/2) eu + C = (1/2) e2x + C.
Is the Derivative that Integral of e come the 2x the Same?
No, the derivative of e2x is 2e2x whereas the integral the e2x is e2x/2 + C. I.e.,d/dx (e2x) = e2x∫ e2x dx = e2x/2 + C
How to find the Integral of e come the 2x through Differentiation?
We know that the derivative of e2x is 2e2x. I.e.,
d/dx (e2x) = 2e2x
d/dx (e2x/2) = e2x
By acquisition integral on both sides,
∫ d/dx (e2x/2) dx = ∫ e2x dx
The integral and d/dx obtain canceled with each various other on the left. So we will be left with
e2x/2 = ∫ e2x dx.
See more: In The Aeroplane Over The Sea Chords By Neutral Milk Hotel, Neutral Milk Hotel
Since we usually include an integration constant C because that every unknown integral,
∫ e2x dx = e2x/2 + C.
What is the Integral that e come the 2x + 1?
To discover the integral ∫ e2x+1 dx, assume the 2x+1 = u. Climate 2 dx = du. From this, we have, dx = du/2. The integral becomes: