Step-by-Step Guide to Finding the Integral of Tan(x) | Substitution Method

∫tanxdx

To find the integral of tan(x), we can use a technique called substitution

To find the integral of tan(x), we can use a technique called substitution. Let’s break down the process step by step:

Step 1: Identify a substitution
We need to identify a substitution in order to simplify the integrand. One common choice for this particular integral is to let u = cos(x). This substitution will allow us to express tan(x) in terms of u.

Step 2: Differentiate the substitution
Once we have our substitution, we need to differentiate it to find du in terms of dx. In this case, if u = cos(x), then du/dx = -sin(x).

Step 3: Rewrite the integral
Now, we need to rewrite the integral in terms of our substitution. The original integral becomes:
∫tan(x) dx = ∫(sin(x)/cos(x)) dx

Using our substitution u = cos(x), we can rewrite the integral in terms of u:
∫(sin(x)/cos(x)) dx = ∫(sin(x)/(u)) (-du)

Step 4: Evaluate the integral
Now, we can simplify the integral and evaluate it in terms of u:
∫(sin(x)/(u)) (-du) = -∫(sin(x)/u) du

The integral -∫(sin(x)/u) du can be evaluated using standard techniques, such as u-substitution or integration by parts. The result is:
-∫(sin(x)/u) du = -ln|u| + C

Step 5: Substitute back
Finally, we substitute back our original substitution u = cos(x) into the result:
-ln|u| + C = -ln|cos(x)| + C

So, the integral of tan(x) is -ln|cos(x)| + C, where C is the constant of integration.

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