Step-by-Step Guide to Finding the Integral of tan(x) with Respect to x

∫ tan x dx

To find the integral of tan(x) with respect to x, we will use a technique called integration by substitution

To find the integral of tan(x) with respect to x, we will use a technique called integration by substitution. Here’s the step-by-step process:

Step 1: Identify a suitable substitution.
We can choose to let u = sin(x), which implies du = cos(x)dx. Using the trigonometric identity tan(x) = sin(x) / cos(x), we can rewrite the integral in terms of u as:
∫ tan(x) dx = ∫ (sin(x) / cos(x)) dx = ∫ (u / cos^2(x)) du.

Step 2: Rewrite the integral using the new variable.
Now that we have expressed the integral in terms of u, we need to substitute the du with its equivalent in terms of dx. Remember that du = cos(x)dx. We can rewrite the integral as:
∫ (u / cos^2(x)) du = ∫ (u / cos^2(x)) (cos(x)dx).

Step 3: Simplify the integral.
We can combine the terms involving u and cos(x) to simplify the integral further:
∫ (u / cos^2(x)) (cos(x)dx) = ∫ (u / cos(x)) du = ∫ usec(x) du.

Step 4: Integrate the simplified integral.
Now it’s a matter of integrating the function usec(x) with respect to u:
∫ usec(x) du = (1/2)u^2sec(x) + C,
where C is the constant of integration.

Step 5: Convert back to the original variable.
Finally, we convert the solution back to the original variable x using the substitution u = sin(x):
(1/2)u^2sec(x) + C = (1/2)sin^2(x)sec(x) + C.

Therefore, the integral of tan(x) with respect to x is (1/2)sin^2(x)sec(x) + C, where C represents the constant of integration.

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