integral of e^x(sqrt(1+e^(2x)))dx
∫ e^x √(1+e^(2x)) dx
Note that if u = e^x, you have
∫ u√(1+u^2) du
Now, if v = 1+u^2, dv = 2u du, and you have
1/2 ∫ √v dv
That I think you can handle, eh?
wouldn't it be the integral of sqrt(1+u^2)du instead of integral u(sqrt (1+u^2)du because the du takes care of the outside u?
you are correct. My bad. Good catch.
To find the integral of e^x(sqrt(1+e^(2x)))dx, we can use a technique called u-substitution. Let's go through the steps:
Step 1: Choose a suitable substitution.
In this case, let u = 1 + e^(2x). We choose this substitution because the derivative of u with respect to x is present in the integrand.
Step 2: Find du/dx and solve for dx.
Differentiating both sides of the equation u = 1 + e^(2x) with respect to x, we get du/dx = 2e^(2x).
Solving this equation for dx, we have dx = (1/2e^(2x))du.
Step 3: Substitute the variables and rewrite the integral.
Using the substitution u = 1 + e^(2x), we substitute u for 1 + e^(2x) and dx for (1/2e^(2x))du in the original integral:
∫ e^x(sqrt(1+e^(2x)))dx = ∫ e^x(sqrt(1+u))(1/2e^(2x))du
Simplifying the expression, we get:
∫ (1/2)sqrt(1+u) du
Step 4: Evaluate the integral with respect to u.
Now, we integrate the expression with respect to u.
∫ (1/2)sqrt(1+u) du = (1/2)∫ sqrt(1+u) du
To solve this integral, we can use the substitution v = 1 + u. The derivative of v with respect to u is dv/du = 1. Solving this equation for du, we have du = dv.
Substituting, we get:
∫ (1/2)√v dv
Step 5: Evaluate the integral and substitute back for u.
Integrating the expression with respect to v, we obtain:
(1/2) * (2/3)(v^(3/2)) + C = (1/3)v^(3/2) + C = (1/3)(1 + u)^(3/2) + C
Finally, substituting back for u, we have:
∫ e^x(sqrt(1+e^(2x)))dx = (1/3)(1 + 1 + e^(2x))^(3/2) + C
= (1/3)(2 + e^(2x))^(3/2) + C
Therefore, the integral of e^x(sqrt(1+e^(2x)))dx is (1/3)(2 + e^(2x))^(3/2) + C, where C is the constant of integration.