Find dx in a Complex Integration Problem: Anti-Derivative of Cos(2x)

[Whatupdoc]

how would i find the anti-derv. of (cos(2x)). I am really confused with sub.

ok i'll use u-du

u = 2x
du = 2 dx
dx = 1/2 du

k so... cos(u)*dx

since the dx is there, you plug in the dx that i found right? so... sin(2x)1/2

ok so that's a simple integration problem, can someone give me a harder problem and show me how to find "dx", that's what I am really confused about.

 

[plover]

So if in general:

u = f(x)
du = f'(x) dx
dx = du/f'(x)​

Find the anti-derivative of:

cos(cos(x)) ⋅ sin(x)​

and use:

u = cos(x)​

as a substitution.

 

[M98Ranger]

To find dx in a complex integration problem, we need to use substitution. In this case, we are given the function cos(2x) and we want to find its anti-derivative.

First, we let u = 2x and du = 2dx. This is because the derivative of 2x is 2, which is the coefficient of x in our function.

Next, we solve for dx by dividing both sides by 2: dx = 1/2 du.

Now, we can substitute this value for dx in our original function: cos(2x)dx = cos(u)(1/2 du).

From here, we can integrate the function as usual, treating u as our variable and using the power rule. The result will be the anti-derivative of cos(2x) in terms of u.

Finally, we can substitute back in for u to get the anti-derivative in terms of x. In this case, the anti-derivative will be sin(2x) + C.

To demonstrate this process with a harder problem, let's find the anti-derivative of e^(3x).

Again, we start by letting u = 3x and du = 3dx. Solving for dx, we get dx = 1/3 du.

Substituting this value for dx in our original function, we get e^(3x)dx = e^u(1/3 du).

Integrating this function with respect to u, we get the anti-derivative 1/3 e^u + C.

Finally, substituting back in for u, we get the anti-derivative in terms of x: 1/3 e^(3x) + C.

In summary, to find dx in a complex integration problem, we use substitution to solve for dx in terms of du. Then, we can integrate the function with respect to du and substitute back in for our original variable to get the final anti-derivative.