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Chapter 4: Problem 20

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

Short Answer

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

Step by step solution

01

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鈥檚 still a missing secant.
02

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 \(\int \sec(x/2)[\sec(x/2) + \tan(x/2)] / (\sec(x/2) + \tan(x/2)) dx\).
03

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 \(\int \frac{t}{2} dt\).
04

Integrate

Now, the integral is in simple form and can be easily resolved. Calculating the integral of \(t/2\) with respect to \(t\) gives \(t^2/4 + C\).
05

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 \(\sec^2(x/2)/4 + \sec(x/2)\tan(x/2)/2 + C\).

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