Antiderivative Calculator
Find the indefinite integral ∫f(x)dx for any function — polynomials, trig, exponentials, logarithms, and fractions — with complete step-by-step working showing every antiderivative rule applied and automatic verification.
⚠️ Use * for multiplication, ^ for powers. Use log(x) for natural log. For 1/x² write 1/x^2 or x^(-2).
Antiderivative (Indefinite Integral)
For any exponent n ≠ −1: increase the exponent by 1, then divide by the new exponent. The power rule for antiderivatives is the most important and most-used integration rule.
Pattern: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
Examples: ∫x dx = x²/2 + C | ∫x³ dx = x⁴/4 + C | ∫x⁻² dx = −1/x + C
Why n ≠ −1? When n = −1: xⁿ⁺¹/(n+1) = x⁰/0 — division by zero. Use the natural log rule for that case.
Constants factor out of integrals — integrate the function first, then multiply by the constant.
Example: ∫7x² dx = 7·∫x² dx = 7·(x³/3) = 7x³/3 + C
Split sums and differences into separate integrals — handle each term independently using the appropriate rule.
Example: ∫(x² + 3x) dx = x³/3 + 3x²/2 + C
The antiderivative of a constant k is kx. Special case: ∫1 dx = x + C (antiderivative of 1).
Examples: ∫1 dx = x + C | ∫5 dx = 5x + C | ∫π dx = πx + C
The special case for n = −1 where the power rule fails. The absolute value |x| is essential — it covers both x > 0 and x < 0.
For x > 0: ∫(1/x) dx = ln(x) + C For x < 0: ∫(1/x) dx = ln(−x) + C Combined: ln|x| + C
Proof: d/dx(ln|x|) = 1/x ✓
The exponential function eˣ is its own antiderivative — the only elementary function with this remarkable property.
∫eˣ dx = eˣ + C | ∫e^(kx) dx = e^(kx)/k + C | ∫aˣ dx = aˣ/ln(a) + C
Examples: ∫eˣ dx = eˣ + C | ∫2ˣ dx = 2ˣ/ln(2) + C
All six standard trigonometric antiderivative rules:
∫sin(x) dx = −cos(x) + C | ∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C | ∫csc²(x) dx = −cot(x) + C
∫sec(x)tan(x) dx = sec(x) + C | ∫csc(x)cot(x) dx = −csc(x) + C
∫1/(1+x²) dx = arctan(x) + C — memorise this one!
∫1/√(1−x²) dx = arcsin(x) + C (valid for |x| < 1)
∫1/(a²+x²) dx = (1/a)·arctan(x/a) + C
Use when the integrand contains f(g(x))·g'(x) — a composite function multiplied by the derivative of the inner function. Let u = g(x), du = g'(x)dx, integrate in u, substitute back.
Example: ∫2x·sin(x²) dx → Let u = x², du = 2x dx → ∫sin(u) du = −cos(u) + C = −cos(x²) + C
For products of functions. Choose u (to differentiate) and dv (to integrate), then apply: ∫u·dv = u·v − ∫v·du
Example: ∫x·eˣ dx → u=x, dv=eˣdx → v=eˣ, du=dx → xeˣ − ∫eˣ dx = xeˣ − eˣ + C = eˣ(x−1) + C
Reference only — requires problem-specific setup beyond direct rule application.
Click any row to load that function into the calculator instantly. Search by typing a function name.
| f(x) | ∫ f(x) dx | Notes |
|---|---|---|
| 1 | x + C | Antiderivative of 1 |
| x | x²/2 + C | Antiderivative of x |
| x² | x³/3 + C | |
| x³ | x⁴/4 + C | Antiderivative of x³ |
| xⁿ | xⁿ⁺¹/(n+1) + C | n ≠ −1; power rule |
| 1/x | ln|x| + C | n=−1 special case |
| 1/x² | −1/x + C | Antiderivative of 1/x² |
| 1/x³ | −1/(2x²) + C | Antiderivative of 1/x³ |
| √x | (2/3)x^(3/2) + C | x^(1/2) |
| 1/√x | 2√x + C | x^(−1/2) |
| 2x | x² + C | Antiderivative of 2x |
| 2x² | (2/3)x³ + C | Antiderivative of 2x² |
| x⁻² | −1/x + C | Same as 1/x² |
| f(x) | ∫ f(x) dx | Notes |
|---|---|---|
| eˣ | eˣ + C | Antiderivative of eˣ |
| e^(kx) | e^(kx)/k + C | k is constant |
| 2ˣ | 2ˣ/ln(2) + C | aˣ/ln(a) + C |
| ln(x) | x·ln(x) − x + C | Integration by parts |
| 1/x | ln|x| + C | Natural log rule |
| f(x) | ∫ f(x) dx | Notes |
|---|---|---|
| sin(x) | −cos(x) + C | Antiderivative of sin(x) |
| cos(x) | sin(x) + C | Antiderivative of cos(x) |
| tan(x) | −ln|cos(x)| + C | = ln|sec(x)| + C |
| sec²(x) | tan(x) + C | |
| csc²(x) | −cot(x) + C | |
| sin²(x) | x/2 − sin(2x)/4 + C | Half-angle formula |
| cos²(x) | x/2 + sin(2x)/4 + C | Half-angle formula |
| f(x) | ∫ f(x) dx | Notes |
|---|---|---|
| 1/(1+x²) | arctan(x) + C | Antiderivative of 1/(1+x²) |
| 1/√(1−x²) | arcsin(x) + C | |x| < 1 |
| −1/√(1−x²) | arccos(x) + C |
What Is an Antiderivative — Definition and Notation
This antiderivative calculator finds the indefinite integral of any function — polynomials, trig, exponentials, logarithms, and fractions — showing step-by-step working with every antiderivative rule applied and automatic verification by differentiating the result back to the original function.
An antiderivative (also called an indefinite integral or primitive function) of f(x) is any function F(x) such that F'(x) = f(x). The standard notation is:
Each part of the notation ∫ f(x) dx = F(x) + C carries meaning:
- ∫ = the integral sign (an elongated S from Latin "summa")
- f(x) = the integrand — the function being integrated
- dx = indicates integration with respect to x
- F(x) = the antiderivative found
- C = the constant of integration — any real number
Why Is There a + C?
Because the derivative of any constant is 0, there are infinitely many antiderivatives — all differing only by a constant. All three of x³/3, x³/3 + 7, and x³/3 − 42 are valid antiderivatives of x², since differentiating any of them gives x². The "+ C" accounts for all of them at once.
Antiderivative vs Definite Integral
- An antiderivative (indefinite integral): ∫f(x)dx = F(x) + C — gives a family of functions
- A definite integral: ∫ₐᵇf(x)dx = F(b) − F(a) — gives a specific number (area under curve)
- The Fundamental Theorem of Calculus connects them: if F(x) = ∫f(x)dx, then ∫ₐᵇf(x)dx = F(b) − F(a)
Example: Using the Antiderivative for a Definite Integral
- Antiderivative: ∫x² dx = x³/3 + C
- Evaluate ∫₀³ x² dx = [3³/3 + C] − [0³/3 + C] = 9 − 0 = 9 (C cancels)
Antiderivative Rules — All Integration Rules Explained
These ten antiderivative rules cover every function encountered in standard calculus. Each rule is the reverse of a differentiation rule.
Power Rule (most important): ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1
The power rule for antiderivatives: increase the exponent by 1, then divide by the new exponent. The power rule applies to all powers except n = −1. Apply the power rule for antiderivatives to every polynomial term.
| Integral | n | n+1 | Result |
|---|---|---|---|
| ∫x dx | 1 | 2 | x²/2 + C |
| ∫x² dx | 2 | 3 | x³/3 + C |
| ∫x³ dx | 3 | 4 | x⁴/4 + C |
| ∫x⁻² dx = ∫1/x² dx | −2 | −1 | −1/x + C |
| ∫x⁻³ dx = ∫1/x³ dx | −3 | −2 | −1/(2x²) + C |
| ∫√x dx = ∫x^(1/2) dx | 1/2 | 3/2 | (2/3)x^(3/2) + C |
Why n ≠ −1? When n = −1: xⁿ⁺¹/(n+1) = x⁰/0 — division by zero is undefined. The special case is ∫x⁻¹ dx = ∫(1/x) dx = ln|x| + C.
Example 1 — Antiderivative of x: ∫x dx
- Apply power rule: n = 1, n+1 = 2
- x^(1+1)/(1+1) = x²/2
- ∫x dx = x²/2 + C
Example 2 — Antiderivative of 2x: ∫2x dx
- Factor out constant 2: 2∫x dx
- 2 × x²/2 = x²
- ∫2x dx = x² + C
- Verification: d/dx(x²) = 2x ✓
Example 3 — Antiderivative of 1/x²: ∫(1/x²) dx
- Rewrite: ∫x⁻² dx
- Power rule: n = −2, n+1 = −1
- x⁻²⁺¹/(−2+1) = x⁻¹/(−1) = −1/x
- ∫(1/x²) dx = −1/x + C
- Verification: d/dx(−1/x) = d/dx(−x⁻¹) = x⁻² = 1/x² ✓
Example 4 — Antiderivative of 2x²: ∫2x² dx
- Factor out 2: 2∫x² dx
- 2 × x³/3 = 2x³/3
- ∫2x² dx = (2/3)x³ + C
Example 5 — Antiderivative of a Polynomial: ∫(4x³ − 6x + 2) dx
- Sum rule: ∫4x³ dx − ∫6x dx + ∫2 dx
- Power rule each term: 4·(x⁴/4) − 6·(x²/2) + 2x = x⁴ − 3x² + 2x
- ∫(4x³ − 6x + 2) dx = x⁴ − 3x² + 2x + C
How to Find an Antiderivative — Step-by-Step Method
Follow this five-step process to find any antiderivative reliably and without errors:
- Step 1 — Identify the form: Is it a power (power rule)? Is it 1/x (natural log)? Is it eˣ (exponential)? Is it trig? Is it a sum? Factor constants out first.
- Step 2 — Apply the rule: Write the formula explicitly, then substitute.
- Step 3 — Simplify: Combine like terms, simplify fractions.
- Step 4 — Add + C: Always append the constant of integration. Never omit it.
- Step 5 — Verify: Differentiate your answer. If you get back the original integrand, the antiderivative is correct.
Example 1 — Polynomial with Negative Exponents: f(x) = 3x² − 2/x² + 5
- Rewrite: 3x² − 2x⁻² + 5
- Split: ∫3x² dx − ∫2x⁻² dx + ∫5 dx
- Power rule each: 3x³/3 − 2x⁻¹/(−1) + 5x = x³ + 2/x + 5x
- Add + C: x³ + 2/x + 5x + C
- Verify: d/dx(x³ + 2x⁻¹ + 5x) = 3x² − 2x⁻² + 5 = 3x² − 2/x² + 5 ✓
Example 2 — Trig: f(x) = 3sin(x) − 2cos(x)
- Split: 3∫sin(x) dx − 2∫cos(x) dx
- Apply trig rules: 3·(−cos(x)) − 2·sin(x) = −3cos(x) − 2sin(x)
- Add + C: −3cos(x) − 2sin(x) + C
- Verify: d/dx(−3cos(x) − 2sin(x)) = 3sin(x) − 2cos(x) ✓
Example 3 — Exponential + Power: f(x) = 4eˣ + 3x²
- Split: 4∫eˣ dx + 3∫x² dx
- Apply rules: 4eˣ + 3·(x³/3) = 4eˣ + x³
- Add + C: 4eˣ + x³ + C
- Verify: d/dx(4eˣ + x³) = 4eˣ + 3x² ✓
Special Antiderivatives — ln|x|, arctan(x), and More
These five antiderivatives are the most commonly searched and do not follow the basic power rule. Memorise all five.
1. Antiderivative of 1/x: ∫(1/x) dx = ln|x| + C
The absolute value |x| is essential — ln is undefined for negative x. For x > 0: ln(x) + C. For x < 0: ln(−x) + C. Combined: ln|x| + C covers both domains. Proof: d/dx(ln|x|) = 1/x ✓
2. Antiderivative of 1/(1+x²): ∫(1/(1+x²)) dx = arctan(x) + C
One of the most important inverse trig antiderivatives — memorise it. Proof: d/dx(arctan(x)) = 1/(1+x²) ✓
3. Antiderivative of 1/√(1−x²): ∫(1/√(1−x²)) dx = arcsin(x) + C
Valid for −1 < x < 1. Proof: d/dx(arcsin(x)) = 1/√(1−x²) ✓
4. Antiderivative of eˣ: ∫eˣ dx = eˣ + C
The exponential function eˣ is its own antiderivative — uniquely remarkable. Proof: d/dx(eˣ) = eˣ ✓
5. Antiderivative of ln(x): ∫ln(x) dx = x·ln(x) − x + C
Requires integration by parts. Proof: d/dx(x·ln(x) − x) = ln(x) + x·(1/x) − 1 = ln(x) + 1 − 1 = ln(x) ✓
Antiderivative vs Derivative — The Fundamental Theorem of Calculus
Differentiation and integration are inverse operations. The Fundamental Theorem of Calculus is the cornerstone connecting them.
| Function f(x) | Derivative f'(x) | Antiderivative ∫f(x)dx |
|---|---|---|
| x³ | 3x² | x⁴/4 + C |
| sin(x) | cos(x) | −cos(x) + C |
| eˣ | eˣ | eˣ + C |
| ln(x) | 1/x | x·ln(x)−x + C |
| arctan(x) | 1/(1+x²) | — (not elementary) |
| x² | 2x | x³/3 + C |
The Fundamental Theorem of Calculus:
Part 1: If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x)
Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x)
This theorem proves that antiderivatives are the key to computing definite integrals — the areas we see in physics, engineering, and data science.
Applying the Fundamental Theorem: ∫₁⁴ (3x²) dx
- Find antiderivative: ∫3x² dx → F(x) = x³ (omit + C since it cancels)
- Apply Part 2: F(4) − F(1) = 4³ − 1³ = 64 − 1 = 63
U-Substitution — Integrating Composite Functions
Use u-substitution when the integrand contains f(g(x))·g'(x) — a composite function multiplied by the derivative of its inner part. It is the reverse of the chain rule.
Method: (1) Let u = g(x). (2) Compute du = g'(x)dx. (3) Rewrite ∫ entirely in u. (4) Integrate. (5) Substitute back g(x). (6) Add + C.
Example 1: ∫2x·cos(x²) dx
- Let u = x², du = 2x dx
- ∫cos(u) du = sin(u) + C
- Substitute back: sin(x²) + C
- Verify: d/dx(sin(x²)) = cos(x²)·2x ✓
Example 2: ∫x·eˣ² dx
- Let u = x², du = 2x dx → x dx = du/2
- ∫eᵘ · (du/2) = (1/2)eᵘ + C
- Substitute back: (1/2)eˣ² + C
- Verify: d/dx((1/2)eˣ²) = x·eˣ² ✓
Example 3: ∫3x²·√(x³+1) dx
- Let u = x³+1, du = 3x² dx
- ∫√u du = (2/3)u^(3/2) + C
- Substitute back: (2/3)(x³+1)^(3/2) + C
- Verify: d/dx = 3x²·√(x³+1) ✓
Common Mistakes When Finding Antiderivatives
Mistake 1 — Forgetting + C
- ❌ Wrong: ∫x² dx = x³/3
- ✅ Correct: ∫x² dx = x³/3 + C
- Without + C you have one specific antiderivative, not the full family.
Mistake 2 — Applying Power Rule to 1/x
- ❌ Wrong: ∫(1/x) dx = x⁰/0 = undefined
- ✅ Correct: ∫(1/x) dx = ln|x| + C — memorise this special case!
Mistake 3 — Forgetting to Divide by the New Exponent
- ❌ Wrong: ∫x³ dx = x⁴ + C
- ✅ Correct: ∫x³ dx = x⁴/4 + C — the power rule says increase exponent AND divide
Mistake 4 — Wrong Chain Rule Reversal
- ❌ Wrong: ∫sin(3x) dx = −cos(3x) + C
- ✅ Correct: ∫sin(3x) dx = −cos(3x)/3 + C (divide by derivative of inner function)
- Verify: d/dx(−cos(3x)/3) = sin(3x)·3/3 = sin(3x) ✓
Mistake 5 — Not Rewriting Before Integrating
- ❌ Wrong attempt: ∫(1/x²) dx using ln rule
- ✅ Correct: Rewrite first as ∫x⁻² dx, then power rule: x⁻¹/(−1) = −1/x + C
Worked Examples — Full Solutions
1. ∫(x³ − 4x + 3) dx
- Sum rule: ∫x³ dx − 4∫x dx + 3∫1 dx
- Power rule each: x⁴/4 − 4·(x²/2) + 3x = x⁴/4 − 2x² + 3x
- Answer: x⁴/4 − 2x² + 3x + C
- Verify: d/dx = x³ − 4x + 3 ✓
2. ∫(1/x² + 1/x³) dx
- Rewrite: ∫x⁻² dx + ∫x⁻³ dx
- Power rule: x⁻¹/(−1) + x⁻²/(−2) = −1/x − 1/(2x²)
- Answer: −1/x − 1/(2x²) + C
- Verify: d/dx(−x⁻¹ − (1/2)x⁻²) = x⁻² + x⁻³ ✓
3. ∫(√x + 1/√x) dx
- Rewrite: ∫x^(1/2) dx + ∫x^(−1/2) dx
- Power rule: (2/3)x^(3/2) + 2x^(1/2)
- Answer: (2/3)x^(3/2) + 2√x + C
4. ∫(2eˣ − 3sin(x)) dx
- Split: 2∫eˣ dx − 3∫sin(x) dx
- Rules: 2eˣ − 3·(−cos(x)) = 2eˣ + 3cos(x)
- Answer: 2eˣ + 3cos(x) + C
- Verify: d/dx = 2eˣ − 3sin(x) ✓
5. ∫(5cos(x) + 2sec²(x)) dx
- Trig rules: 5sin(x) + 2tan(x)
- Answer: 5sin(x) + 2tan(x) + C
6. ∫(1/(1+x²) + x²) dx
- Inverse trig + power rule: arctan(x) + x³/3
- Answer: arctan(x) + x³/3 + C
7. ∫(4x⁷ − 3x⁴ + x − 7) dx
- Power rule each term: 4·(x⁸/8) − 3·(x⁵/5) + x²/2 − 7x
- Answer: x⁸/2 − 3x⁵/5 + x²/2 − 7x + C
8. ∫(eˣ + 1/x) dx
- Exponential + natural log: eˣ + ln|x|
- Answer: eˣ + ln|x| + C
9. ∫sin²(x) dx — Half-Angle Formula
- Identity: sin²(x) = (1 − cos(2x))/2
- ∫(1/2 − cos(2x)/2) dx = x/2 − sin(2x)/4
- Answer: x/2 − sin(2x)/4 + C
10. ∫(x + 1)² dx — Expand First
- Expand: (x+1)² = x² + 2x + 1
- ∫(x² + 2x + 1) dx = x³/3 + x² + x
- Answer: x³/3 + x² + x + C
- Verify: d/dx = x² + 2x + 1 = (x+1)² ✓