Discover How to Integrate the Function tan(x) Using the Powerful Technique of Integration by Substitution

tanx dx

To integrate the function tan(x) with respect to x, we can use a technique called integration by substitution

To integrate the function tan(x) with respect to x, we can use a technique called integration by substitution.

Let’s start by applying a substitution. Let u = tan(x), which implies du = sec^2(x) dx. We can rearrange this equation to give dx = du / sec^2(x).

Now, we substitute the values of du and dx back into the integral:

∫ tan(x) dx = ∫ tan(x) (du / sec^2(x))

Next, we simplify the integrand by canceling sec^2(x):

= ∫ tan(x) du

The integral on the right side represents the antiderivative of tan(x) with respect to u. Recall that the derivative of tan(x) is sec^2(x). Therefore, the antiderivative of tan(x) with respect to u is ln|sec(x)| + C, where C is the constant of integration.

Finally, we substitute the value of u back into the equation:

= ln|sec(x)| + C

So, the integral of tan(x) with respect to x is ln|sec(x)| + C.

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