Step-by-Step Guide: Integrating tan(x)dx Using Substitution Technique

tan(x)dx

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

To integrate the function tan(x)dx, we can use a technique called substitution. Let’s go through the steps:

Step 1: Choose a suitable substitution.
In this case, we can let u = tan(x). This allows us to express the derivative of u with respect to x as du/dx = sec^2(x). Rearranging, we have dx = du / sec^2(x).

Step 2: Rewrite the integral in terms of the substitution.
The original integral is now rewritten as ∫ tan(x) dx = ∫ u * (du / sec^2(x)).

Step 3: Simplify the integral using the substitution.
Since dx = du / sec^2(x), we substitute it into the integral to get:
∫ u * (du / sec^2(x)) = ∫ u * (du / (1 + tan^2(x))).

Step 4: Simplify the integral further.
Using the trigonometric identity 1 + tan^2(x) = sec^2(x), we can simplify the integral as:
∫ u * (du / (1 + tan^2(x))) = ∫ u * du / sec^2(x).

Step 5: Simplify the integral once more.
Since sec^2(x) is equal to 1/cos^2(x), we can rewrite the integral as:
∫ u * du / sec^2(x) = ∫ u * cos^2(x) * du.

Step 6: Integrate the simplified integral.
Now we integrate the simplified expression:
∫ u * cos^2(x) * du = ∫ u * (1 – sin^2(x)) * du.

Expanding, we get: ∫ u – u * sin^2(x) * du.

The integral of u with respect to u is simply u^2/2, and the integral of sin^2(x) with respect to x is (2x – sin(2x))/4 or x/2 – (sin(2x))/4.

Thus, the final result is:
∫ u * cos^2(x) * du = (u^2/2) – (u * x/2) + (u * sin(2x))/4.

Step 7: Convert back to the original variable.
Substituting back u = tan(x), we get the final answer as:
∫ tan(x) dx = (tan^2(x)/2) – (tan(x) * x/2) + (tan(x) * sin(2x))/4 + C,
where C is the constant of integration.

So, the integral of tan(x) dx is given by (tan^2(x)/2) – (tan(x) * x/2) + (tan(x) * sin(2x))/4 + C.

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