Evaluate the integral.

{eq}\int_{\pi/4}^{\pi/3} \frac{\ln(\tan x)}{\sin x \cos x} \, \mathrm{d}x {/eq}

Question:

Integration Using Substitution:

Let's say we have a function {eq}\displaystyle I = \int g(f(x)) \cdot f'(x) dx {/eq}.

If we need to evaluate the value of the integral, we can simplify it first using the method of substitution. For example if {eq}\displaystyle u = f(x) {/eq} then, {eq}\displaystyle du = f'(x) dx {/eq}. Notice that the function {eq}\displaystyle f'(x) dx {/eq} can be replaced by {eq}\displaystyle du {/eq}.

Hence by making these substitutions we get,

{eq}\displaystyle I = \int g(f(x)) f'(x) dx = \int g(u) du {/eq}

Answer and Explanation: 1