Mastering Integration by Substitution: A Step-by-Step Guide to Integrating tan(x) dx

tanx dx

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

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

Let’s make a substitution:
Let u = tan(x)
Differentiating both sides with respect to x, we get:
du = sec^2(x) dx

Rearranging the equation, we can write dx = du / sec^2(x)

Substituting these values into the original integral, we have:
∫ tan(x) dx = ∫ u * (du / sec^2(x))

But sec^2(x) is equal to 1 + tan^2(x), so we can rewrite the integral as:
∫ u * (du / (1 + u^2))

Now, we can integrate the function with respect to u using the formula for the integral of 1/(1 + u^2):
∫ (u / (1 + u^2)) du = (1/2) ln|1 + u^2| + C

Finally, we substitute back u = tan(x):
∫ tan(x) dx = (1/2) ln|1 + tan^2(x)| + C

This is the final antiderivative (or indefinite integral) of tan(x) dx. Note that C is the constant of integration, which accounts for the fact that there are infinitely many antiderivatives for any given function.

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