Integrate [sec(x) tan(x)] / [3 sec(x) + 5] dx

Solution:

We will be using the concepts of differentiation and integration to solve this.

Let's solve this step by step.

Given that, ∫ [sec(x) tan(x)] / [3 sec(x) + 5] dx

Let z = [3 sec(x) + 5]

Differentiate z with respect to x:

dz/dx = 3sec(x) tan(x)

sec(x) tan(x) dx = dz / [3]

Substitute the value of dx in the integration:

I = ∫ [sec(x) tan(x)] / [3 sec(x) + 5] dx = 1/3 ∫ 1/z dz

I = ln(z) / 3

Substitue the value of z above:

I = ln(3 sec(x) + 5) / 3

Hence, ∫ [sec(x) tan(x)] / [3 sec(x) + 5] dx = ln(3 sec(x) + 5) / 3

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Integrate [sec(x) tan(x)] / [3 sec(x) + 5] dx

Summary:

Integration of [sec(x) tan(x)] / [3 sec(x) + 5] dx is equal to ln(3 sec(x) + 5) / 3