Example Data Table
| Function |
a |
b |
x |
Bounds |
Main rule |
| arctan(ax + b) |
2 |
1 |
0.5 |
0 to 1 |
u arctan(u) - 0.5 ln(1 + u²) |
| arcsin(ax + b) |
0.5 |
0 |
1 |
-1 to 1 |
u arcsin(u) + sqrt(1 - u²) |
| arcsec(ax + b) |
1 |
2 |
1 |
0 to 2 |
u arcsec(u) - sgn(u) ln|u + sqrt(u² - 1)| |
Formula Used
The calculator uses the substitution u = ax + b. Since du/dx = a, each base antiderivative is divided by a.
arcsin: ∫ arcsin(u) dx = [u arcsin(u) + sqrt(1 - u²)] / a + C
arccos: ∫ arccos(u) dx = [u arccos(u) - sqrt(1 - u²)] / a + C
arctan: ∫ arctan(u) dx = [u arctan(u) - 0.5 ln(1 + u²)] / a + C
arccot: ∫ arccot(u) dx = [u arccot(u) + 0.5 ln(1 + u²)] / a + C
arcsec: ∫ arcsec(u) dx = [u arcsec(u) - sgn(u) ln|u + sqrt(u² - 1)|] / a + C
arccsc: ∫ arccsc(u) dx = [u arccsc(u) + sgn(u) ln|u + sqrt(u² - 1)|] / a + C
How to Use This Calculator
- Select the inverse trig function.
- Enter coefficient a and shift b for u = ax + b.
- Enter an x value for antiderivative evaluation.
- Enter lower and upper bounds for the definite integral.
- Add the constant C if you want an indefinite value.
- Choose decimal precision.
- Press Calculate.
- Use CSV or PDF export after the result appears.
Inverse Trig Antiderivatives Guide
Why These Integrals Matter
Inverse trigonometric antiderivatives appear in calculus, geometry, physics, and signal work. They connect slopes, angles, ratios, and curved motion. Many students memorize final rules, yet miss the structure behind them. This calculator focuses on that structure. It handles a linear inner expression, so the usual chain rule factor is included.
The Core Idea
Most inverse trig integrals use integration by parts. You treat the inverse trig term as the first factor. You treat one as the second factor. The remaining integral usually becomes a square root or logarithm term. When the inside is ax plus b, substitution adds the divisor a. That is why the coefficient must not be zero.
Definite and Indefinite Results
An indefinite antiderivative gives a family of functions. The constant C represents every vertical shift. A definite integral uses two bounds instead. The constant cancels, so only the change in the antiderivative matters. This page shows both ideas together. It also evaluates the antiderivative at a selected x value.
Domain Awareness
Inverse sine and inverse cosine require the inner value to stay between negative one and one. Inverse secant and inverse cosecant need an absolute inner value at least one. Inverse tangent and inverse cotangent accept every real inner value. Bounds outside a required domain cannot produce a real result. The calculator warns you when that happens.
Practical Uses
These forms help in arc length problems, angle models, navigation, optics, and engineering checks. They also support classroom verification. A teacher can build examples quickly. A student can compare manual steps with computed values. The export buttons save the result for later review.
Good Study Habits
Always identify the inner expression first. Confirm its coefficient. Check the domain before trusting a decimal answer. Then compare the symbolic result with the evaluated value. If the function uses secant or cosecant, watch the absolute value and logarithm term carefully. Small domain mistakes can change the answer. Use this calculator as a guide, not a replacement for understanding.
Checking Results
Different antiderivative forms can look different but still be equivalent. Different constants and logarithm identities may change the appearance. Differentiate the final expression when unsure. The derivative should match the original integrand.
FAQs
What does this calculator integrate?
It integrates inverse trig functions with a linear inner expression, written as ax + b. It supports arcsin, arccos, arctan, arccot, arcsec, and arccsc.
Why must coefficient a not be zero?
The formulas use u = ax + b. If a is zero, the substitution factor fails. The expression becomes constant, so this calculator asks for a nonzero coefficient.
Can it calculate definite integrals?
Yes. Enter lower and upper bounds. The calculator evaluates F(upper) - F(lower) after checking the real domain for the selected inverse function.
Does the constant C affect definite integrals?
No. The constant cancels during subtraction. It only affects the displayed antiderivative value at the selected x input.
Why do arcsin and arccos show domain warnings?
They need the inner value to stay between -1 and 1. If ax + b leaves that range, a real result is not available.
Why do arcsec and arccsc need special checks?
They require |ax + b| to be at least 1. Their antiderivatives also include logarithmic terms with square roots.
Can I export the result?
Yes. After calculating, use the CSV or PDF button. The export includes inputs, formula, domain note, and computed values.
Is this useful for homework?
Yes. It shows formulas, substitution steps, domain notes, and numerical checks. You should still verify steps with your course method.