Find the Antiderivative Calculator

Calculator Inputs

Function to integrate

Examples: 3*x^2 + 4*x - 5, sin(x), cos(2*x), exp(x), 1/x, sqrt(x).

Variable

Constant of integration

Check point for original function

Lower bound

Upper bound

Output actions

Formula Used

The calculator uses linearity first: ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx.

For powers, it applies ∫a*x^n dx = a*x^(n+1)/(n+1), where n is not -1.

For reciprocal terms, it applies ∫a/x dx = a*ln|x|.

For trigonometric forms, it uses ∫sin(kx) dx = -cos(kx)/k and ∫cos(kx) dx = sin(kx)/k.

For exponential forms, it uses ∫exp(kx) dx = exp(kx)/k.

How to Use This Calculator

Enter the function with clear multiplication signs.
Choose the variable, such as x, t, or y.
Enter a constant symbol, or keep C.
Add optional bounds to estimate a definite value.
Press Calculate to view the result above the form.
Use CSV or PDF buttons to save the current result.

Example Data Table

Function	Variable	Expected Antiderivative	Rule
3x^2 + 4x - 5	x	x^3 + 2x^2 - 5x + C	Power and constant rules
sin(x)	x	-cos(x) + C	Sine rule
cos(2*x)	x	0.5sin(2x) + C	Linear inner rule
1/x	x	ln\|x\| + C	Log rule
exp(3*x)	x	0.33333333exp(3x) + C	Exponential rule

Understanding Antiderivatives

Antiderivatives turn a known rate into a family of original functions. They are also called indefinite integrals. This calculator helps students explore that idea in a practical way. It reads common algebraic and trigonometric patterns. It then applies standard rules and displays a clean combined answer.

Why Steps Matter

The main rule is linearity. Each separated term is integrated on its own. Then the pieces are added again. This makes long expressions easier to study. A polynomial term uses the power rule. A sine term becomes a negative cosine term. A cosine term becomes a sine term. An exponential term keeps its shape, but it is divided by the inner rate.

Writing Clear Input

Good notation matters. Use x^2 for powers. Use sin(x), cos(x), exp(x), sqrt(x), and ln(x) for common functions. Add multiplication signs where possible. For example, write 3*x^2 rather than 3x^2. Clear input reduces mistakes and improves the explanation.

Review and Export

The calculator also supports review features. You can add a constant of integration. You can enter bounds to estimate a definite value numerically. You can enter a test point for checking. The CSV export is useful for spreadsheets. The PDF export is helpful for saving a short report.

Learning Notes

An antiderivative is not always unique. All correct answers differ by a constant. That is why the final result includes C. In applied problems, an initial condition can define a specific constant. Without that condition, the answer remains a family of curves.

This tool is meant for learning and planning. It handles many routine forms, but very complex symbolic expressions may need separate algebra. When a term is not recognized, it is still shown in the result as an unevaluated integral. That makes the limitation clear. You can simplify the expression, split it into smaller parts, and try again. Use the formula section to compare each step with standard integration rules.

For best results, review the sample table before entering your own work. It shows how different patterns change. Try one idea at a time, then add more terms. This habit makes errors easier to find. It also helps you understand why each rule was selected for the final answer with confidence and better problem solving each day.

FAQs

What is an antiderivative?

An antiderivative is a function whose derivative equals the original function. It represents a family of functions because any constant can be added without changing the derivative.

Why does the answer include C?

C represents the constant of integration. Indefinite integrals need it because many functions can share the same derivative while differing by a constant.

Which syntax should I use?

Use x^2 for powers and add multiplication signs. Write sin(x), cos(x), exp(x), ln(x), and sqrt(x) for common functions.

Can it solve every integral?

No. It handles many common patterns. If a term is not recognized, the calculator keeps it as an unevaluated integral for manual review.

How are bounds used?

Bounds are used to estimate a definite integral numerically. Leave them blank when you only need the indefinite antiderivative family.

What does the check point do?

The check point evaluates the original function at one selected value. It helps you inspect the input and confirm numeric behavior.

Why should I export CSV?

CSV export saves the expression, answer, and step table. It is useful when you want to compare results in a spreadsheet.

Why should I export PDF?

PDF export creates a compact report. It is helpful for sharing the expression, final answer, and main integration steps.