Evaluating the Integral of tan(x) | Integration by Substitution

int tanx

The expression “int tanx” refers to the integral of the tangent function, which can be written as ∫ tan(x) dx

The expression “int tanx” refers to the integral of the tangent function, which can be written as ∫ tan(x) dx.

To evaluate this integral, we can use a technique called integration by substitution. We start by letting u = tan(x), so du/dx = sec^2(x) and dx = du/sec^2(x).

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

Next, we need to express the sec^2(x) term in terms of u. Recall that sec^2(x) = 1 + tan^2(x). Since u = tan(x), we have u^2 + 1 = sec^2(x).

Now, we can rewrite the integral as:
∫ u * (du/sec^2(x)) = ∫ u * (du/(u^2 + 1))

To solve this integral, we can use a substitution again. Let v = u^2 + 1, so dv = 2u du. Solving for du, we have du = dv/(2u).

Substituting these expressions into the integral, we get:
∫ (u * (1/(u^2 + 1))) du = ∫ (1/2) * (dv/v)

Now, this integral is straightforward to solve as it is the natural logarithm function:
∫ (1/2) * (dv/v) = (1/2) * ln(|v|) + C

Remembering that v = u^2 + 1, we can substitute back:
(1/2) * ln(|u^2 + 1|) + C

Finally, substituting u back into the expression:
(1/2) * ln(|tan(x)^2 + 1|) + C

Therefore, the integral of tan(x) is (1/2) * ln(|tan(x)^2 + 1|) + C, where C is the constant of integration.

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