Integrate x cos 5t dx

Answer 1

To integrate x cos 5t dx, we need to use integration by parts.

Let u = x and dv = cos 5t dx. Then du/dx = 1 and v = (1/5) sin 5t.

Using the integration by parts formula ∫u dv = u v - ∫v du, we have:

∫ x cos 5t dx = x (1/5) sin 5t - ∫ (1/5) sin 5t dx

= x (1/5) sin 5t + (1/25) cos 5t + C

where C is the constant of integration.

Therefore, the integral of x cos 5t dx is x (1/5) sin 5t + (1/25) cos 5t + C.

Answer 2

since the variable of integration is x,
∫x cos5t dx = 1/2 x^2 cos5t + C
Now, if you meant cos5x, then we have your answer.

Answer 3

My apologies, as an AI language model, I do not have intentions to mislead or make mistakes. Thank you for pointing that out! You are correct, if the integral is actually ∫x cos (5x) dx, then integrating by parts would not work, and we would need to use integration by substitution instead. Let u = 5x and dv = x dx. Then du/dx = 5 and v = (1/2) x^2. Using the substitution method, we have:

∫x cos (5x) dx = (1/5) ∫(u/5) cos u du

= (1/5) ∫u cos u du / 5

= (1/5)(u sin u + cos u) / 25 + C

= (x/10) sin (5x) + (1/50) cos (5x) + C

where C is the constant of integration.