Integrate x cos 5t dx
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.
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.
To integrate the expression ∫x cos(5t) dx, we can follow these steps:
Step 1: Integration by parts
Start by using the integration by parts formula: ∫u dv = uv - ∫v du. Let's choose u = x and dv = cos(5t) dx.
Step 2: Differentiate u
Differentiating u = x with respect to x, we have du = dx.
Step 3: Integrate dv
Integrating dv = cos(5t) dx with respect to x:
∫cos(5t) dx = (1/5)(∫cos(5t) d(5t))
= (1/5)(∫cos(5t) dt)
Step 4: Integrate ∫cos(5t) dt
Integrating ∫cos(5t) dt gives us:
(1/5)(∫cos(5t) dt) = (1/5)(∫cos(u) du)
= (1/5)(sin(u)) + C
where C is the constant of integration, and u = 5t.
Step 5: Apply the integration by parts formula
Using the integration by parts formula: ∫u dv = uv - ∫v du, we have:
∫x cos(5t) dx = (1/5)(x sin(5t)) - ∫(1/5)(sin(5t)) dx
Step 6: Simplify the integral
The integral term can be simplified as:
∫(1/5)(sin(5t)) dx = (-1/25)(cos(5t)) + C
where C is the constant of integration.
Step 7: Final Answer
Putting everything together, we have:
∫x cos(5t) dx = (1/5)(x sin(5t)) - (1/25)(cos(5t)) + C
where C is the constant of integration.
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.