Integral of (x \sec x \tan x) – A Step‑by‑Step Guide
The integral
[
\int x,\sec x,\tan x ,dx
]
is a classic example that combines a product of a polynomial and a trigonometric function with a derivative–substitution trick. In this article we will walk through the solution in detail, explore the underlying ideas, and finish with a few related examples and FAQs.
Some disagree here. Fair enough.
Introduction
When encountering an integral that mixes algebraic and trigonometric parts, the first instinct is often to look for a substitution that simplifies the expression. In the case of (\int x,\sec x,\tan x ,dx), the natural candidate is the derivative of (\sec x), which is (\sec x \tan x). This observation leads directly to an elegant solution via integration by parts and a simple substitution Surprisingly effective..
Step 1: Recognize the Derivative Inside the Integrand
The derivative of (\sec x) is [ \frac{d}{dx}\sec x = \sec x \tan x. In practice, ] Thus, the integrand can be rewritten as [ x,\sec x,\tan x = x \cdot \frac{d}{dx}(\sec x). ] This form suggests the use of integration by parts, where one factor is differentiated and the other is integrated That's the part that actually makes a difference. Nothing fancy..
Step 2: Apply Integration by Parts
Recall the formula: [ \int u,dv = u,v - \int v,du. ] Choose [ u = x \quad \text{and} \quad dv = \sec x \tan x ,dx. ] Then [ du = dx \quad \text{and} \quad v = \sec x ] (because (\int \sec x \tan x ,dx = \sec x)).
Plugging into the formula gives: [ \int x,\sec x,\tan x ,dx = x \sec x - \int \sec x ,dx. ]
Step 3: Evaluate the Remaining Integral
The remaining integral is the standard one: [ \int \sec x ,dx = \ln!Still, \left|\sec x + \tan x\right| + C. ] So we obtain [ \int x,\sec x,\tan x ,dx = x \sec x - \ln!\left|\sec x + \tan x\right| + C.
Final Result
[ \boxed{\displaystyle \int x,\sec x,\tan x ,dx = x \sec x - \ln!\left|\sec x + \tan x\right| + C} ]
where (C) is the constant of integration.
Scientific Explanation
Why Integration by Parts Works Here
Integration by parts is essentially the reverse of the product rule for differentiation. When we set (u = x), we are exploiting the fact that differentiating a linear term simplifies it to a constant, which removes the variable from that part of the integrand. The remaining part, (\sec x \tan x), is precisely the derivative of (\sec x), making the integration straightforward It's one of those things that adds up. Worth knowing..
The Role of the Logarithm
The integral of (\sec x) is known to involve a logarithm because the antiderivative of (\sec x) cannot be expressed in elementary algebraic terms alone. Its derivation uses a clever multiplication by (\frac{\sec x - \tan x}{\sec x - \tan x}) to transform the integrand into a form that yields a natural logarithm after simplification.
Alternative Approach: Substitution First
Some students prefer to start with a substitution before applying integration by parts. Let’s see how that works.
-
Set (t = \sec x).
Then (dt = \sec x \tan x ,dx), or (dx = \frac{dt}{t \sqrt{t^2-1}}) (since (\tan^2 x = \sec^2 x - 1)) Most people skip this — try not to. And it works.. -
Rewrite the integral: [ \int x,\sec x,\tan x ,dx = \int x,dt. ] But now (x) must be expressed in terms of (t). From (t = \sec x), we have (x = \sec^{-1} t), which is not convenient Most people skip this — try not to..
Thus, the substitution route becomes messy because (x) cannot be cleanly expressed in terms of (t). This illustrates why the integration‑by‑parts route is preferable for this particular integral That's the part that actually makes a difference..
Related Integrals
| Integral | Technique | Result |
|---|---|---|
| (\displaystyle \int \sec^3 x ,dx) | Integration by parts + reduction | (\frac{1}{2}\sec x\tan x + \frac{1}{2}\ln |
| (\displaystyle \int x \cos x ,dx) | Integration by parts | (x\sin x + \cos x + C) |
| (\displaystyle \int x \sec^2 x ,dx) | Integration by parts | (x\tan x - \ln |
These examples share the same pattern: a polynomial factor multiplied by a trigonometric function whose derivative appears elsewhere in the integrand Not complicated — just consistent..
FAQ
1. What if the integral were (\int x,\csc x,\cot x ,dx)?
Answer:
Since (\frac{d}{dx}\csc x = -\csc x \cot x), we can set (u=x) and (dv = -\csc x \cot x,dx). The result is
[
\int x,\csc x,\cot x ,dx = -x\csc x + \ln|\csc x + \cot x| + C.
]
2. Can we differentiate the result to verify it?
Answer:
Yes. Differentiate (x \sec x - \ln|\sec x + \tan x|):
[
\frac{d}{dx}\bigl(x \sec x\bigr) = \sec x + x \sec x \tan x,
]
[
\frac{d}{dx}\bigl(\ln|\sec x + \tan x|\bigr) = \frac{\sec x \tan x + \sec^2 x}{\sec x + \tan x} = \sec x.
]
Subtracting gives (x \sec x \tan x), confirming the integration.
3. What if the integrand were (x^2 \sec x \tan x)?
Answer:
Apply integration by parts twice:
- Let (u=x^2), (dv=\sec x \tan x,dx).
(du=2x,dx), (v=\sec x).
Result: (x^2\sec x - \int 2x\sec x,dx). - For (\int 2x\sec x,dx), set (u=2x), (dv=\sec x,dx).
(du=2,dx), (v=\ln|\sec x + \tan x|).
Result: (2x\ln|\sec x + \tan x| - 2\int \ln|\sec x + \tan x|,dx).
The remaining integral is more involved and may require advanced techniques or numerical methods.
Conclusion
The integral (\int x,\sec x,\tan x ,dx) demonstrates how a keen observation—recognizing the derivative of (\sec x)—can simplify a seemingly complex problem. On top of that, by combining integration by parts with the known antiderivative of (\sec x), we arrive at a clean, closed‑form solution. Now, this method generalizes to many similar integrals where a polynomial multiplies a trigonometric function whose derivative is present elsewhere in the integrand. Armed with this strategy, you can tackle a wide range of integrals that appear in calculus courses and beyond That's the part that actually makes a difference. Simple as that..
