Calculator
Graph
Example Data Table
| Example | a | b | n | Unit | Limits | Expected numeric value | Reason |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 1 | Radians | 0 → π/4 | ≈ 0.346574 | Equals ln(√2) from −ln|cos x|. |
| 2 | 2 | 0 | 1 | Radians | 0 → π/8 | ≈ 0.173287 | Half-angle range with scale 1/a. |
| 3 | 1 | 0 | 2 | Radians | 0 → π/4 | ≈ 0.214602 | Uses tan²x = sec²x − 1. |
Formula Used
Start with the substitution u = a·x + b (or u = (a·x + b)·π/180 in degrees).
Then du = a·dx, so dx = du/a (or dx = 180·du/(a·π)).
For integer powers, the calculator uses the identity tan²(u) = sec²(u) − 1
to build a reduction rule:
∫ tan(a·x+b)^n dx = (scale) · Jₙ(u) + C, where scale = 1/a.
How to Use
- Enter a and b to define tan(a·x + b).
- Set n to integrate tan(a·x + b)n.
- Choose Radians or Degrees to match your input.
- Select a preferred output form for the base case when n = 1.
- Use Definite mode to estimate a value between limits.
- If limits cross an asymptote, enable the principal value split.
- Click Calculate. Use the download buttons for files.
Article
1) What this calculator solves
This tool computes integrals built from tangent, especially ∫ tan(a·x + b)n dx. It helps when you change the frequency (a) or shift the angle (b). It also warns when your chosen range nears vertical asymptotes where tan becomes undefined.
2) Inputs, limits, and units
Enter a, b, and an integer power n from 1 to 30. Choose radians or degrees to match your problem. In degrees, angles are converted with π/180 and the substitution scale updates automatically. You can also set the variable symbol (x, t, etc.) for cleaner output. Switch the n=1 form between −ln|cos(u)| and ln|sec(u)| to match your textbook’s preferred answer format for grading and review.
3) Base identity for n = 1
For n = 1 the key rule is ∫ tan(u) du = −ln|cos(u)| + C, equivalent to ln|sec(u)| + C. Here u equals a·x + b in radians, or (a·x + b)·π/180 in degrees. If a = 0, tan(b) is constant (when defined), so the integral becomes tan(b)n·x + C.
4) Power reduction for integer n
For n ≥ 2 it uses a reduction that steps down by two: Jn(u) = tan(u)n−1/(n−1) − Jn−2(u), with J0(u)=u and J1(u)=−ln|cos(u)| (or ln|sec(u)|). So n=10 reduces to 8, 6, 4, 2, 0, keeping expressions compact.
5) Definite mode and asymptote checks
Definite mode estimates ∫xLxU tan(a·x+b)n dx using adaptive Simpson integration. If an asymptote lies between limits, the integral may diverge. Enabling principal value splits the interval around detected asymptotes and integrates only on safe sub-ranges.
6) Graph and example numbers
The plot shows tan(a·x+b) and caps visible values at ±10 for readability near steep regions. In definite mode, a filled trace highlights tan(a·x+b)n inside your limits. A standard check is ∫0π/4 tan(x) dx ≈ 0.346574, matching ln(√2).
7) Downloads and practical workflow
Use CSV export to store inputs, the symbolic result, and numeric estimates for spreadsheets or worksheets. The PDF export creates a one-page summary for sharing. For demonstrations, increase plot points for smoother curves, and keep limits away from values where a·x+b approaches π/2 + kπ (or 90° + 180k).
FAQs
Q1: What expression does the calculator integrate?
A: It targets ∫ tan(a·x + b)n dx for integer n (1–30). With n=1 it returns the standard logarithmic antiderivative, and for n≥2 it applies a reduction that steps down by two powers.
Q2: Why are there two forms for the n=1 answer?
A: −ln|cos(u)| and ln|sec(u)| differ only by a constant because sec(u)=1/cos(u). Either is correct; choose the one that matches your notes, grading key, or textbook style.
Q3: How does “Degrees” change the result?
A: When angles are in degrees, u=(a·x+b)·π/180, so dx introduces a factor 180/(a·π). The calculator applies this automatically to keep the antiderivative consistent with degree-based inputs.
Q4: What happens if my limits cross an asymptote?
A: Tangent is undefined where a·x+b hits π/2+kπ (or 90°+180k). The definite integral may diverge. The tool detects interior asymptotes and suggests enabling principal value to split the interval safely.
Q5: What does “principal value” mean here?
A: It integrates on sub-intervals that stop just short of each asymptote, then sums those parts. This can produce a meaningful symmetric cancellation in special cases, but it does not “fix” truly divergent integrals.
Q6: What do the CSV and PDF downloads include?
A: Both exports include your inputs, the symbolic antiderivative, and any numeric definite estimate plus notes. CSV is easy for spreadsheets and worksheets; the PDF is a compact one-page record for sharing or archiving.