Problem 20 - Essential Calculus

Question

Find the indefinite integral. $$ \int \sec \frac{x}{2} d x $$

Step-by-Step Solution

Answer

The result of the integral is \(\sec^2(x/2)/4 + \sec(x/2)\tan(x/2)/2 + C\)

Step 1: Recognize Form

The given function is not directly integrable. To find the integral, recognize that the derivative of \(\tan(x)\) is \(\sec^2(x)\). \(\sec(x/2)\) looks like derivative of \(\tan(x/2)\), but instead of \(\sec^2(x/2)\), there’s still a missing secant.

Step 2: Use a Clever Trick

It's often a clever trick to multiply and divide an expression by the same term. Here, multiply and divide by \(\sec(x/2) + \tan(x/2)\). Thus, the given integral becomes \(\int \sec(x/2)[\sec(x/2) + \tan(x/2)] / (\sec(x/2) + \tan(x/2)) dx\).

Step 3: Trigonometric Substitution

Let's represent \(\sec(x/2) + \tan(x/2)\) as some variable \(t\). The derivative \(\frac{1}{2}[\sec^2(x/2) + \sec(x/2)\tan(x/2)]dx\) is same as \(dt\). Our integral now becomes \(\int \frac{t}{2} dt\).

Step 4: Integrate

Now, the integral is in simple form and can be easily resolved. Calculating the integral of \(t/2\) with respect to \(t\) gives \(t^2/4 + C\).

Step 5: Back-Substitution

The last step is to replace the variable \(t\) with the original trigonometric terms. So the answer is \([(\sec(x/2) + \tan(x/2))^2]/4 + C\) which simplifies to \(\sec^2(x/2)/4 + \sec(x/2)\tan(x/2)/2 + C\).