Th integral of tan(3x)dx
tan (3x) = sin(3x)/cos(3x)
so ∫tan(3x) dx
= ∫sin(3x)/cos(3x) dx
= -(1/3) ln(cos(3x)) + c
To find the integral of tan(3x)dx, you can use a technique called substitution. Let's go through the steps:
Step 1: Start by letting u = 3x. This will be our substitution variable.
Step 2: Calculate du/dx. Since u = 3x, du/dx = 3.
Step 3: Rearrange the equation du/dx = 3 to solve for dx. Dividing both sides by 3, we get dx = du/3.
Step 4: Now substitute u and dx back into the integral. The integral of tan(3x)dx becomes the integral of tan(u) * (du/3).
Step 5: Simplify the expression. Since dx = du/3, we can pull out the constant factor of 1/3 from the integral.
Step 6: Recall that the integral of tan(u)du is equal to ln|sec(u)| + C, where C is the constant of integration.
Step 7: Bring back the constant factor of 1/3. The final answer is (1/3) * ln|sec(u)| + C.
Step 8: Replace u with 3x. The final answer is (1/3) * ln|sec(3x)| + C.
So, the integral of tan(3x)dx is (1/3) * ln|sec(3x)| + C.