Handle radical integrals with substitution choices and steps. Export tables, review curves, and verify values. Practice difficult expressions using clear inputs and reliable outputs.
Domain note: Use x ≥ a for families containing sqrt(x^2 - a^2). Use |x| ≤ a for families containing sqrt(a^2 - x^2).
The graph shows the radical term that drives the substitution pattern for the selected integral family.
| Integral Family | a | x | Substitution | Antiderivative Value |
|---|---|---|---|---|
| ∫ dx / sqrt(a^2 - x^2) | 3 | 1 | x = a sin(θ) | 0.339837 |
| ∫ sqrt(a^2 - x^2) dx | 4 | 2 | x = a sin(θ) | 7.652892 |
| ∫ dx / sqrt(a^2 + x^2) | 5 | 3 | x = a tan(θ) | 0.568825 |
| ∫ sqrt(x^2 - a^2) dx | 3 | 5 | x = a sec(θ) | 5.056245 |
Trig substitution works by matching the radical to a trigonometric identity. The three core identities are 1 - sin2(θ) = cos2(θ), 1 + tan2(θ) = sec2(θ), and sec2(θ) - 1 = tan2(θ).
Use these matching substitutions:
After substitution, rewrite the radical and dx, simplify the trigonometric integral, integrate, and convert the result back to x. This calculator also evaluates common antiderivative families directly, making it useful for checking steps, definite integrals, and transformed expressions.
Integral trig substitution is a standard method for evaluating integrals that contain radicals such as sqrt(a2 - x2), sqrt(a2 + x2), and sqrt(x2 - a2). These patterns are difficult to integrate directly because the variable appears inside a square root in a way that does not simplify well with ordinary algebra alone.
The method replaces x with a trigonometric expression involving an angle θ. That replacement converts the radical into a simpler product of trig functions. Once the radical and dx are rewritten, the original integral becomes a new trigonometric integral that is often easier to simplify and solve. After integrating in θ, the answer is converted back into x by using inverse trigonometric relationships or right-triangle identities.
This calculator is useful for students, teachers, and anyone reviewing advanced integration techniques. It highlights the correct substitution pattern, the transformed integral, the angle relation, and the resulting antiderivative formula. It also supports direct evaluation at a chosen point and definite integral estimation through closed-form antiderivative differences. The included graph helps visualize the radical term, which often improves intuition about domain restrictions and substitution choice.
Because domain matters, the calculator checks whether the chosen x-values fit the selected radical family. That makes it practical for homework checks, quick revision, and verifying manual algebra during multistep integration problems.
Trig substitution replaces x with a trigonometric form so a radical simplifies through a known identity. It is commonly used for square roots involving a2 - x2, a2 + x2, or x2 - a2.
Use x = a sin(θ) when the radical contains sqrt(a2 - x2). The identity 1 - sin2(θ) = cos2(θ) converts the radical into a cos(θ) expression.
Use x = a tan(θ) for radicals shaped like sqrt(a2 + x2). The identity 1 + tan2(θ) = sec2(θ) makes the square root simplify into a sec(θ) term.
Use x = a sec(θ) for radicals of the form sqrt(x2 - a2). This uses sec2(θ) - 1 = tan2(θ), which turns the radical into a tan(θ) expression.
Yes. Enter both lower and upper bounds. The calculator applies the matching antiderivative formula at both endpoints and subtracts the values, provided the selected bounds stay inside the valid domain.
Domain limits keep the radical real and the inverse trig or inverse hyperbolic step valid. For example, sqrt(a2 - x2) requires |x| ≤ a, while sqrt(x2 - a2) needs x ≥ a in this tool.
The graph shows the radical term connected to your chosen integral family. It helps you see shape, valid input range, and how the substitution is tied to the behavior of the square root.
Yes. You can download result data or example data as CSV files. You can also print the result, graph, or examples section and save that print view as a PDF.