Question

In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int \sec 2 x \tan 2 x d x, u=2 x$$

Short answer

The integral \(\int \sec 2x \tan 2x dx\) is equal to \(\frac{1}{2} \ln|\sec 2x + \tan 2x| + C\), where C is the constant of integration.

Step-by-step solution

Apply the Substitution

Using the substitution \(u = 2x\), differentiate both sides to get \(du = 2dx\), or \(dx = du/2\). Also, rewrite \(\sec 2x \tan 2x dx\) as \(\sec u \tan u (du/2)\).

Simplify the Integral Using the Substitution

The integral becomes \(\int \sec u \tan u (du/2) = \frac{1}{2} \int \sec u \tan u du\). This is a standard integral one may remember it as the integral of \(\sec u\) , which is \(\ln|\sec u + \tan u|\).

Evaluate the Integral

So the integral becomes \(\frac{1}{2} \ln|\sec u + \tan u| + C\), where C is the constant of integration.

Back Substitute 'u' with '2x'

Substitute \(u\) back in terms of \(x\), so the answer becomes \(\frac{1}{2} \ln|\sec 2x + \tan 2x| + C\).

Verify the Answer by Differentiation

By differentiating \(\frac{1}{2} \ln|\sec 2x + \tan 2x| + C\) with respect to \(x\), if the result is \(\sec 2x \tan 2x\), this confirms that the integral was evaluated correctly.