Integration Calculator — Integrals, Antiderivatives & Rules

Integration is the reverse of differentiation — and it also measures area under curves. These two ideas (antiderivative and area) are connected by the Fundamental Theorem of Calculus, one of the most remarkable results in all of mathematics. Check your integrals with the CalcHub Integration Calculator.

Indefinite vs. Definite Integrals

Indefinite integral: Finds the family of antiderivatives. $$\int f(x)\, dx = F(x) + C$$ Definite integral: Computes the net area between the curve and x-axis from a to b. $$\int_a^b f(x)\, dx = F(b) - F(a)$$

The constant C disappears in definite integrals because it cancels out.

The Fundamental Theorem of Calculus

If F'(x) = f(x), then:
$$\int_a^b f(x)\, dx = F(b) - F(a)$$

This is why antiderivatives matter — finding one gives you definite integrals for free.

Basic Integration Rules

Power Rule

$$\int x^n\, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$

∫x⁵ dx = x⁶/6 + C
∫x⁻² dx = x⁻¹/(−1) + C = −1/x + C
∫√x dx = ∫x^(1/2) dx = x^(3/2)/(3/2) + C = (2/3)x^(3/2) + C

Exception: ∫(1/x) dx = ln|x| + C

Constant Multiple and Sum Rules

$$\int cf(x)\, dx = c\int f(x)\, dx$$ $$\int [f(x) \pm g(x)]\, dx = \int f(x)\, dx \pm \int g(x)\, dx$$

Common Integrals Reference Table

Function	Integral
xⁿ (n≠−1)	xⁿ⁺¹/(n+1) + C
1/x	ln\	x\	+ C
eˣ	eˣ + C
aˣ	aˣ/ln(a) + C
sin x	−cos x + C
cos x	sin x + C
tan x	ln\	sec x\	+ C
sec²x	tan x + C
sec x tan x	sec x + C
1/√(1−x²)	arcsin x + C
1/(1+x²)	arctan x + C
1/(a²+x²)	(1/a) arctan(x/a) + C

Worked Example: Definite Integral

Evaluate ∫₁³ (3x² − 2x + 1) dx

Step 1 — Find antiderivative:
F(x) = x³ − x² + x

Step 2 — Apply bounds:
F(3) = 27 − 9 + 3 = 21
F(1) = 1 − 1 + 1 = 1

Result: 21 − 1 = 20

Integration by Substitution (u-Substitution)

Used when the integrand contains a function and its derivative.

Pattern: ∫f(g(x))·g'(x) dx Method: Let u = g(x), du = g'(x) dx Example: ∫2x·(x²+1)⁵ dx Let u = x² + 1, du = 2x dx ∫u⁵ du = u⁶/6 + C = (x²+1)⁶/6 + C Example: ∫sin(3x) dx Let u = 3x, du = 3 dx → dx = du/3 ∫sin(u) · du/3 = −cos(u)/3 + C = −cos(3x)/3 + C

Geometric Interpretation

The definite integral gives the signed area between the curve and x-axis:

• Area above x-axis contributes positively


• Area below x-axis contributes negatively

For total (unsigned) area when the curve crosses the x-axis, split the integral at the crossing points and add absolute values.

Applications:

• Area between two curves: ∫[f(x) − g(x)] dx
• Volume of revolution (disk method): π∫[f(x)]² dx
• Average value of a function: (1/(b−a)) ∫ₐᵇ f(x) dx

Why does ∫(1/x) dx = ln|x| and not x⁰/0?

The power rule breaks down at n = −1 because it would give x⁰/0 = 1/0 — undefined. The correct antiderivative of 1/x is the natural logarithm, which can be shown by differentiating ln|x| using the chain rule and definition of e.

What is integration by parts?

It's the reverse of the product rule: ∫u dv = uv − ∫v du. Useful for products like ∫x·eˣ dx or ∫x·sin(x) dx. A good mnemonic for choosing u is "LIATE": Logarithm, Inverse trig, Algebraic, Trig, Exponential — pick the first matching type as u.

When can't an integral be expressed in closed form?

Many integrals don't have a closed-form antiderivative — for example, ∫eˣ²dx or ∫sin(x)/x dx. These are handled numerically (Gaussian quadrature, Simpson's rule) or expressed using special functions (like the error function erf(x)).

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