Integral Calculator

Integration

Integral Type

Function f(x)

∫ x^2 + 3x + 1 dx

Variable:

Quick Examples

Antiderivative

x^3/3 + 3 · (x^2/2) + 1x + C

Sum/Difference Rule

Function Graph

Graph of f(x) = x^2 + 3x + 1

f(x)

Step-by-Step Solution

See how the integral is computed

1Split the integral

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

2Left: Split the integral

∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx

3Left: Left: Power Rule

∫ x^n dx = x^(n+1)/(n+1) + C

n = 2, n+1 = 3

x^3/3 + C

4Left: Right: Factor out constant

∫ 3 · f(x) dx = 3 · ∫ f(x) dx

5Left: Right: Power Rule

∫ x dx = x^(1+1)/(1+1) + C

n = 1

x^2/2 + C

6Left: Right: Multiply by constant

3 · (x^2/2) + C

7Left: Combine results

x^3/3 + 3 · (x^2/2) + C

8Right: Constant Rule

∫ c dx = cx + C

c = 1

1x + C

9Combine results

x^3/3 + 3 · (x^2/2) + 1x + C

Common Integral Formulas

Quick reference table of standard integrals

∫ xⁿ dx

= xⁿ⁺¹/(n+1) + C

(n ≠ -1)

∫ 1/x dx

= ln|x| + C

∫ eˣ dx

= eˣ + C

∫ aˣ dx

= aˣ/ln(a) + C

(a > 0)

∫ sin(x) dx

= -cos(x) + C

∫ cos(x) dx

= sin(x) + C

∫ tan(x) dx

= -ln|cos(x)| + C

∫ sec²(x) dx

= tan(x) + C

∫ csc²(x) dx

= -cot(x) + C

∫ ln(x) dx

= x·ln(x) - x + C

What Is an Integral?

The fundamental concept of calculus that reverses differentiation

An integral computes the accumulation of quantities. In calculus, integration is the reverse process of differentiation. If the derivative tells you the rate of change, the integral tells you the total accumulated change.

There are two types: the indefinite integral (antiderivative) finds a family of functions whose derivative is the given function, while the definite integral computes the net signed area between a function and the x-axis over an interval.

Fundamental Theorem of Calculus

If F'(x) = f(x), then ∫f(x) dx = F(x) + C

∫_a^b f(x) dx = F(b) - F(a)

Common Integration Techniques

Methods for finding antiderivatives of different function types

Power Rule

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

The most fundamental integration rule. Increase the exponent by 1 and divide by the new exponent.

U-Substitution

When the integrand contains a composite function f(g(x)) and its derivative g'(x) appears as a factor, substitute u = g(x) to simplify. This is the reverse of the chain rule.

Example

∫ 2x cos(x²) dx

Let u = x², du = 2x dx

= ∫ cos(u) du = sin(u) + C = sin(x²) + C

Integration by Parts

∫ u dv = uv - ∫ v du

Used when the integrand is a product of two functions. Choose u (to differentiate) and dv (to integrate) using the LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

Definite vs. Indefinite Integrals

Understanding the two types of integrals and when to use each

Indefinite Integral

Finds a family of antiderivatives
Result includes + C (constant)
No bounds of integration

Definite Integral

Computes a numerical value
Result is F(b) - F(a)
Represents net signed area

Common Integration Mistakes to Avoid

Frequent errors and how to prevent them

Forgetting + C

Indefinite integrals always include a constant of integration. Omitting it loses a family of valid solutions.

Wrong power rule for n = -1

The power rule doesn't work for ∫ x⁻¹ dx. This integral equals ln|x| + C, not x⁰/0.

Missing chain rule factor

When integrating sin(3x), the result is -cos(3x)/3, not -cos(3x). Don't forget to divide by the inner derivative.

Treating ∫ as distributive over products

∫ f(x)g(x) dx ≠ (∫ f dx)(∫ g dx). Integration is linear only for sums, not products.

Frequently Asked Questions

Common questions about integrals and integration techniques