how would i find the integral of sin (2t+1)/cos^2(2t+1) dt
To find the integral of sin((2t+1)/cos^2(2t+1)) dt, you can use a substitution method. Here's a step-by-step guide on how to do it:
Step 1: Let u = 2t + 1. We will use this substitution to simplify the integral.
Step 2: Find du/dt by differentiating both sides of the equation u = 2t + 1 with respect to t. This will give you du/dt = 2.
Step 3: Solve the above equation for dt by dividing both sides by 2. You will have dt = du/2.
Step 4: Substitute the value of u and dt obtained from Steps 1 and 3 into the integral:
∫ sin((2t+1)/cos^2(2t+1)) dt = ∫ sin(u/cos^2(u)) (du/2)
Step 5: The expression sin(u/cos^2(u)) can be rewritten as (sin(u)/cos^2(u)) or tan(u)/cos(u). Now, the integral becomes:
(1/2) ∫ tan(u)/cos(u) du
Step 6: The integral of tan(u)/cos(u) can be evaluated by using a substitution. Let v = cos(u), which means dv = -sin(u) du. Rearrange this equation to solve for du: du = -dv/sin(u).
Step 7: Substitute the values of v and du into the integral:
(1/2) ∫(tan(u)/cos(u)) du = (1/2) ∫(-tan(u)/v) dv
Step 8: The integral on the right side is now in a simpler form. It can be solved by using the natural logarithm:
(1/2) ∫(-tan(u)/v) dv = -(1/2) ln|v| + C
Step 9: Substitute the value of v back in terms of u:
(1/2) ln|v| = (1/2) ln|cos(u)|
Step 10: Convert back to the original variable t by substituting the value of u:
(1/2) ln|cos(u)| = (1/2) ln|cos(2t + 1)|
Step 11: Combining all the steps, the final result of the integral is:
∫ sin((2t+1)/cos^2(2t+1)) dt = -(1/2) ln|cos(2t + 1)| + C
where C is the constant of integration.
let u = (2t+1)
du = 2 dt so dt = .5 du
so
.5 integral du sin u/cos^2u
try 1/cos u
d/du (1/cos u) = -sin u/cos^2 u
You can take it from there :)