Mastering the Technique of Integration: Simplifying and integrating the function ∫tan u du using substitution

∫tan u du

To integrate the function ∫tan u du, we can use the technique of substitution

To integrate the function ∫tan u du, we can use the technique of substitution.

Let’s start by simplifying the integral using the trigonometric identity: tan u = sin u / cos u.

∫tan u du = ∫(sin u / cos u) du

Next, let’s make the substitution: let w = cos u. This means that dw = -sin u du.

Now, we can rewrite the integral in terms of w:

∫(sin u / cos u) du = ∫(-1 / w) dw

We also need to find the limits of integration for w. To do this, we can analyze the original variable, u:

When u = 0, w = cos 0 = 1
When u = π/2, w = cos (π/2) = 0

So our new limits of integration for w are from 1 to 0.

Now, let’s continue integrating:

∫(-1 / w) dw = -∫(1 / w) dw

Integrating -1/w is straightforward:

-∫(1 / w) dw = -ln|w| + C

Now, we substitute back for w:

-(-ln|w| + C) = ln|w| + C

Since w = cos u, the final answer is:

ln|cos u| + C

So, the integral of tan u du is ln|cos u| + C, where C is the constant of integration.

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