Solving the Integral of tan(x) using Substitution: Step-by-Step Guide

∫tan(x)dx

To integrate ∫tan(x)dx, we can use a technique called substitution

To integrate ∫tan(x)dx, we can use a technique called substitution.

Let’s substitute u = tan(x), where u is a new variable.

Now, we need to find an expression for dx in terms of du, which is the derivative of u with respect to x.

Differentiating both sides of u = tan(x) with respect to x gives:
du/dx = sec^2(x)

From this, we can solve for dx in terms of du:
dx = du / sec^2(x)

Now, substitute these values into the original integral:
∫tan(x)dx = ∫tan(x) (dx / sec^2(x))

Multiplying and simplifying, we get:
∫tan(x)dx = ∫(tan(x) sec^2(x)) du

We can rewrite tan(x) sec^2(x) as sin(x)/cos^2(x) by using trigonometric identities.

Now, our integral becomes:
∫(sin(x) / cos^2(x)) du

To integrate this, we can further simplify the expression by using another substitution. Let’s substitute v = cos(x), where v is another new variable.

Differentiating both sides of v = cos(x) with respect to x gives:
dv/dx = -sin(x)

Multiplying both sides by -1 and rearranging, we have:
-sin(x) dx = dv

Now, substitute these values into the integral:
∫(sin(x) / cos^2(x)) du = ∫(-1 / v^2) dv

Integrating -1/v^2 with respect to v gives:
∫(-1 / v^2) dv = 1/v

Substituting v back in terms of x, we get:
1/v = 1/cos(x) = sec(x)

Finally, the integral becomes:
∫tan(x)dx = sec(x) + C

Therefore, the solution to ∫tan(x)dx is sec(x) + C, where C represents the constant of integration.

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