int 0^1e^2edx =
This expression is illegible.
Try rewriting to make it clearer.
If you mean
∫[0,1] e^2x dx
then just recall that d/dx (e^u) = e^u du/dx
However, I don't see any x's in your integrand.
To find the value of the integral ∫(0 to 1) e^(2x) dx, you can follow these steps:
Step 1: Identify the integral and its limits:
The given integral is ∫(0 to 1) e^(2x) dx. Here, we are integrating the function e^(2x) with respect to x from x = 0 to x = 1.
Step 2: Solve the integral:
To solve this integral, we will use the power rule for integration, which states that ∫e^(kx) dx = (1/k)e^(kx) + C, where k is a constant and C is the constant of integration.
In this case, the constant is 2, so the integral becomes:
∫e^(2x) dx = (1/2)e^(2x) + C
Step 3: Evaluate the integral using the limits of integration:
To evaluate the integral with the limits from 0 to 1, we substitute the limits in the result obtained in the previous step.
First, substitute x = 1:
(1/2)e^(2*1) = (1/2)e^2
Next, substitute x = 0:
(1/2)e^(2*0) = (1/2)e^0 = (1/2)
Step 4: Calculate the difference between the two evaluated values:
Now, we subtract the value obtained by substituting the lower limit (0) from the value obtained by substituting the upper limit (1) to find the definite integral:
(1/2)e^2 - (1/2) = (e^2 - 1)/2
Therefore, the value of the integral ∫(0 to 1) e^(2x) dx is (e^2 - 1)/2.
To evaluate the integral ∫[0 to 1]e^2x dx, we can use the technique of integration. The integral of e^2x can be found easily by applying the power rule of integration.
The power rule states that the integral of e^kx dx is equal to (1/k) * e^kx + C, where C is the constant of integration.
In this case, we have k = 2, so the integral of e^2x dx can be found as follows:
∫[0 to 1] e^2x dx = (1/2) * e^2x ∣[0 to 1]
To evaluate the integral, let's substitute the upper bound (1) into the expression and then subtract the value of the lower bound (0) from it:
= (1/2) * e^2(1) - (1/2) * e^2(0)
Now, e^2(0) is equal to 1, since any number raised to the power of 0 is equal to 1. Also, e^2(1) is simply e^2.
Therefore, evaluating the expression, we have:
= (1/2) * e^2 - (1/2)
Hence, the value of the integral ∫[0 to 1] e^2x dx is (1/2) * e^2 - (1/2).