who do you solve this?/what is this equal to? I don't understand...
antiderivative [(ln(2+sin(t))] dt
THANK YOU!
To solve for the antiderivative of ln(2+sin(t)), we can use a technique called integration by substitution. Here's how you can solve it step by step:
Step 1: Start by letting u = 2 + sin(t). We will differentiate this expression to find du/dt.
du/dt = d(2 + sin(t))/dt
= 0 + cos(t)
= cos(t)
Step 2: Rearrange the equation from Step 1 to solve for dt.
dt = du / cos(t)
Step 3: Replace ln(2+sin(t)) and dt in the original integral with their respective substitutions.
∫(ln(2+sin(t))) dt = ∫(ln(u)) dt = ∫(ln(u) * dt / cos(t))
Step 4: Substitute all the variables in the integral with their u equivalents.
∫ ln(u) * dt / cos(t) = ∫ ln(u) * (du / cos(t))
Step 5: Simplify the expression by canceling out the common factor cos(t).
∫ ln(u) * (du / cos(t)) = ∫ ln(u) * (du / 1)
= ∫ ln(u) * du
Step 6: At this point, you can integrate ln(u) with respect to u. The antiderivative of ln(u) is u * ln(u) - u + C, where C is the constant of integration.
Therefore, the antiderivative of ln(2+sin(t)) is (2+sin(t)) * ln(2+sin(t)) - (2+sin(t)) + C.
Hope this helps!