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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