A Step-by-Step Guide | Integrating tan(u) du using Substitution

∫ tanu du

To find the integral of tan(u) du, we can use the technique of integration by substitution

To find the integral of tan(u) du, we can use the technique of integration by substitution.

Let’s start by recalling the derivative of the tangent function:

d/dx(tan(x)) = sec^2(x)

Now, let’s make a substitution to simplify the integral.

Let u = x, so du = dx.

The integral becomes:

∫ tan(u) du

Since du = dx, we can rewrite the integral as:

∫ tan(u) du = ∫ tan(u) du

Now, we need to rewrite the integrand in terms of u. We know that:

tan(u) = sin(u) / cos(u)

Replacing tan(u) with sin(u) / cos(u), the integral becomes:

∫ (sin(u) / cos(u)) du

Next, we will make another substitution:

Let v = cos(u), so dv = -sin(u) du.

Rearranging the equation, we can express sin(u) du in terms of dv:

sin(u) du = -dv

Substituting this back into the integral, we get:

∫ (-1/v) dv

Now we can integrate this new expression. The integral becomes:

-∫ (1/v) dv

Integrating -1/v with respect to v gives:

– ln|v| + C

Substituting back v = cos(u), we get:

– ln|cos(u)| + C

Finally, replace u with x (since we made u = x at the beginning), to obtain the final result:

– ln|cos(x)| + C

Therefore, the integral of tan(u) du is – ln|cos(x)| + C, where C is the constant of integration.

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