Calculator
Example data table
These sample rows use degrees for quick reference.
| x (deg) | sec(x) | tan(x) | ln|sec(x)+tan(x)| |
|---|---|---|---|
| 0 | 1.000000 | 0.000000 | 0.000000 |
| 30 | 1.154701 | 0.577350 | 0.549306 |
| 45 | 1.414214 | 1.000000 | 0.881374 |
| 60 | 2.000000 | 1.732051 | 1.316958 |
Formula used
∫ sec(x) dx = ln|sec(x) + tan(x)| + C
This works because the derivative of sec(x) + tan(x) equals sec(x)(sec(x)+tan(x)), which matches the numerator after a helpful multiply-and-divide step.
How to use this calculator
- Select Indefinite for the general antiderivative, or Definite to evaluate bounds.
- Choose Radians or Degrees. Bounds and x follow this unit.
- For definite mode, enter a and b. Avoid intervals crossing where cos(x)=0.
- Adjust decimals, enable step outline, and optionally run a numeric check.
- Press Calculate. Download CSV or PDF from the result panel.
Quick guide: integrating sec(x)
1) Understanding the integral of sec x
The secant function is 1/cos(x), so it grows fast near vertical asymptotes. Its antiderivative looks unusual compared with sine and cosine, but it has a clean logarithmic form that is widely taught in calculus. The tool can also evaluate the formula at x.
2) Where the function is defined
sec(x) is undefined whenever cos(x)=0. In radians, that happens at x = π/2 + kπ. In degrees, it happens at 90° + 180°k. Close to these points, sec(x) and tan(x) become very large.
3) The key substitution idea
A classic trick multiplies the integrand by (sec(x)+tan(x))/(sec(x)+tan(x)). The numerator becomes sec(x)sec(x)+sec(x)tan(x). That matches the derivative of sec(x)+tan(x), which turns the integral into an “u′/u” pattern. Once u = sec(x)+tan(x), the remaining integral is simply ln|u|.
4) The exact closed form result
The calculator uses the identity ∫sec(x)dx = ln|sec(x)+tan(x)| + C. The constant C can be any real value, so the tool lets you label it as C, K, or another symbol. For example, at 0° the value is ln|1+0|=0. At 45° it is ln|1.4142+1|≈0.8814, matching the example table.
5) Checking the answer quickly
You can verify by differentiation. Differentiate ln|sec(x)+tan(x)| and apply the chain rule: (sec(x)tan(x)+sec²(x))/(sec(x)+tan(x)) = sec(x). The absolute value keeps the log argument positive, but the derivative still returns sec(x) wherever the expression is defined. If the tool shows “undefined,” the issue is usually cos(x)=0 or a nonpositive log argument from rounding.
6) Definite integrals and improper cases
For definite mode, the tool evaluates F(b)−F(a) using the same antiderivative. If the interval avoids cos(x)=0, the value is well behaved. If an asymptote lies inside, the integral is improper and should be split at that point, then treated with limits. When you enable the numeric check, Simpson’s rule estimates the area on a continuous interval and reports the absolute difference.
7) Practical uses and tips
Secant integrals appear in trig substitutions, slope and curvature problems, and some optics models. Always confirm your unit selection, because 60 and 60° are very different inputs. Use the step outline to learn the algebraic trick, then reuse it on related forms like ∫sec(x)tan(x)dx. After computing, export CSV or PDF to keep a record of inputs and results.
FAQs
1) What is the integral of sec(x)?
An antiderivative is ln|sec(x)+tan(x)| + C. It differentiates back to sec(x) on intervals where cos(x) is not zero and the logarithm argument stays positive after the absolute value.
2) Why does the answer include a logarithm?
The trick creates a u′/u pattern with u = sec(x)+tan(x). Integrating u′/u gives ln|u|, which becomes ln|sec(x)+tan(x)|.
3) When will the calculator show “undefined”?
It can happen at x values where cos(x)=0, such as 90° or π/2, or when sec(x) and tan(x) overflow near an asymptote. Move x away from those points.
4) Can I compute a definite integral safely?
Yes, if the interval does not cross cos(x)=0. If it does, the integral is improper and should be split at the asymptote and evaluated with limits.
5) Should I use degrees or radians?
Use the unit that matches your inputs. Trig functions change with units, so 60 and 60° are not equivalent. The example table uses degrees for quick checking.
6) What does the numeric verification do?
It estimates ∫sec(x)dx over the bounds using Simpson’s rule, then compares it to F(b)−F(a). This helps catch input mistakes and highlights issues near discontinuities.