Antiderivative Calculator
This antiderivative calculator works an indefinite integral the way a textbook does: the sum rule first, then the power rule, the rule for cos(ax), and ln|x| for 1/x.
Antiderivative
- Read as
- 3x² + 4cos(2x) + 1/x
- Antiderivative
- x³ + 2sin(2x) + ln|x| + C
- Step 1: Sum rule on 3x² + 4cos(2x) + 1/x ∫(f + g) = ∫f + ∫g, one term at a time
- x³ + 2sin(2x) + ln|x|
- Step 2: Constant multiple rule on 3x² ∫c·f = c·∫f
- x³
- Step 3: Power rule on x² ∫xⁿ dx = xⁿ⁺¹/(n + 1), n ≠ -1
- x³/3
- Step 4: Constant multiple rule on 4cos(2x) ∫c·f = c·∫f
- 2sin(2x)
- Step 5: Integral of cos(ax + b) on cos(2x) ∫cos(u) du = sin(u), divided by a for u = ax + b
- sin(2x)/2
- Step 6: Log rule on 1/x ∫1/x dx = ln|x|
- ln|x|
- Check Differentiated, the antiderivative matches the integrand at 24 test points to within 1.0 × 10^-16 relative. The two forms differ only by an identity such as d/dx ln|x| = 1/x.
- d/dx [x³ + 2sin(2x) + ln|x|] = 3x² + 4cos(2x) + 1/x
- Trigonometric functions take their argument in radians.
How the Integral Calculator works
An antiderivative of f is a function F whose derivative is f, and finding one is the same as working out an indefinite integral. The antiderivative calculator finds them by rule, one term at a time, and differentiates every answer back to confirm it before showing it.
The example. For 3x² + 4cos(2x) + 1/x, the sum rule splits it into three integrals. The power rule gives ∫3x² dx = x³. For 4cos(2x), the inside is 2x, so the antiderivative of cos is divided by 2: 4·sin(2x)/2 = 2sin(2x). And 1/x is the one power the power rule cannot handle, since it would divide by zero; its antiderivative is ln|x|. Together: x³ + 2sin(2x) + ln|x| + C.
Why the absolute value. ln(x) is only defined for positive x, but 1/x is defined for negative x too. For x < 0, d/dx ln(-x) = -1/(-x) = 1/x, so ln|x| covers both sides. The calculator drops the bars when the inside can never be negative, which is why ∫2x/(x² + 1) dx comes out as ln(x² + 1).
The rules it knows. Powers of x and of ax + b, including square roots and negative powers. eˣ and other exponentials of ax + b. sin, cos, tan, sec, csc and cot of ax + b, and sec², csc², sin², cos² and tan². 1/(c + kx²), which gives arctan, and 1/√(c - kx²), which gives arcsin. ln of ax + b. And u-substitution whenever the derivative of an inside function appears as a factor: ∫x²(x³ + 1)⁴ dx = (x³ + 1)⁵/15 + C.
What has no answer by rule. Integration by parts (∫x·eˣ dx), partial fractions (∫1/(x² - 1) dx) and trigonometric substitution (∫√(1 - x²) dx) are not attempted, and the calculator says which one an integral looks like it needs. Some functions, such as e^(x²) and sin(x)/x, have no antiderivative that can be written with the usual functions at all, though their definite integrals still have values.
Integral Calculator FAQ
What is an antiderivative?
A function whose derivative is the one you started with. x³ is an antiderivative of 3x², and so is x³ + 4. The general antiderivative, x³ + C, is the indefinite integral, written ∫3x² dx.
What is the antiderivative of 1/x?
ln|x| + C. The power rule xⁿ⁺¹/(n + 1) breaks down at n = -1 because it divides by zero, and the natural logarithm fills the gap. The absolute value makes it work for negative x as well.
How do I check an antiderivative?
Differentiate it. If the derivative is the integrand, the antiderivative is right. d/dx [x³ + 2sin(2x) + ln|x|] = 3x² + 4cos(2x) + 1/x.
Is an indefinite integral the same as an antiderivative?
Effectively, yes. The indefinite integral ∫f(x) dx is the whole family of antiderivatives, F(x) + C, where F is any one of them. The two names are used interchangeably in most courses.