If I’m trying to integrate (-10e^(3x))/(1+e^x)dx, how exactly do I solve? I understand you use u-substitution, right? I think it’s best to substitute u for 1+e^x but I’m not really sure
Nevermind, I think I figured it out.
I hope you got
∫-10(e-1)^3/u du
...
To integrate the expression (-10e^(3x))/(1+e^x)dx, you are correct that u-substitution is a good approach. And yes, you can choose to substitute u for 1+e^x. Let's go through the steps of this integration using u-substitution:
Step 1: Choose u and find du
To start with, you have u = 1 + e^x. Now, differentiate both sides of this equation with respect to x to find du/dx. This will give you:
du/dx = d(1 + e^x)/dx = 0 + d(e^x)/dx = e^x
Step 2: Express dx in terms of du
Rearrange the equation from step 1 to solve for dx:
dx = (1/e^x) * du
Step 3: Substitute the variables in the integral
Now, substitute u and dx into the original integral:
∫ [(-10e^(3x))/(1+e^x)] dx = ∫ [(-10e^(3x))/(u)] [(1/e^x) * du]
Step 4: Simplify the expression
Combine the terms and simplify:
∫ (-10e^(2x)) du
Step 5: Integrate
Now integrate the expression with respect to u, treating x as a constant:
-10 ∫ e^(2x) du = -10 ∫ e^(2x) du
Step 6: Substitute back x for u
Replace u with 1 + e^x:
-10 ∫ e^(2x) du = -10 ∫ e^(2x) d(1 + e^x)
Step 7: Distribute the differential and integrate
Distribute the differential d(1 + e^x):
-10 ∫ e^(2x) d(1 + e^x) = -10 ∫ e^(2x) + e^(3x) dx
Step 8: Integrate the terms separately
Integrate both terms separately:
-10(1/2) ∫ e^(2x) dx - 10(1/3) ∫ e^(3x) dx
Step 9: Simplify and evaluate
Evaluate the integrals:
-5 ∫ e^(2x) dx - (10/3) ∫ e^(3x) dx
The integrals of e^(2x) and e^(3x) can be easily calculated. After evaluating these integrals, you will have your final answer.