Solving Integrals | Integration by Substitution for ∫tan(x)dx

∫tan(x)dx

To evaluate the integral ∫tan(x)dx, we can use a technique called integration by substitution

To evaluate the integral ∫tan(x)dx, we can use a technique called integration by substitution. Let’s go through the steps:

Step 1: Identify a substitution. In this case, a good choice is to let u = tan(x). Then, we can rewrite the entire integral in terms of u.

Step 2: Find the derivative of the substitution. Differentiating both sides of the equation u = tan(x) with respect to x gives du/dx = sec^2(x). Notice that sec^2(x) is the derivative of tan(x).

Step 3: Rewriting the integral. Now, we need to express dx in terms of du to replace it in the integral. Rearranging the equation from Step 2, we have du = sec^2(x) dx.

Step 4: Replacing variables in the integral. We can substitute u for tan(x) and du for sec^2(x) dx in the original integral ∫tan(x)dx to get ∫u du.

Step 5: Evaluating the integral. The integral of u with respect to u is simply (1/2)u^2. Therefore, the antiderivative of tan(x) is (1/2)tan^2(x) + C, where C is the constant of integration.

So, ∫tan(x)dx = (1/2)tan^2(x) + C.

Please note that this result assumes the integration is taken over a domain where tan(x) is defined and continuous.

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