Integral Calculator

What is Integration?

Integration is the reverse process of differentiation. It has two main interpretations:

Geometric: Area under a curve
Physical: Accumulation of quantities over time/space

Two Types:

Indefinite Integral: ∫f(x)dx = F(x) + C (antiderivative + constant)
Definite Integral: ∫[a,b]f(x)dx = F(b) - F(a) (specific numerical value)

Common Integration Rules

Power Rule

∫x^n dx = x^(n+1)/(n+1) + C (for n ≠ -1)

Examples: ∫x² dx = x³/3 + C, ∫x³ dx = x⁴/4 + C

Exponential Functions

∫e^x dx = e^x + C
∫a^x dx = a^x/ln(a) + C

Trigonometric Functions

∫sin(x) dx = -cos(x) + C
∫cos(x) dx = sin(x) + C
∫sec²(x) dx = tan(x) + C

Special Cases

∫(1/x) dx = ln|x| + C
∫(1/(1+x²)) dx = arctan(x) + C

Fundamental Theorem of Calculus

Connects differentiation and integration:

Part 1: If F(x) = ∫[a,x] f(t)dt, then F'(x) = f(x)

Part 2: ∫[a,b] f(x)dx = F(b) - F(a), where F is any antiderivative of f

Examples

Example 1: Indefinite Integral

Q: Find ∫(3x² + 2x)dx

Solution:

∫3x² dx = 3·(x³/3) = x³
∫2x dx = 2·(x²/2) = x²
Answer: x³ + x² + C

Example 2: Definite Integral

Q: Evaluate ∫[0,2] x² dx

Solution:

Find antiderivative: F(x) = x³/3
Apply bounds: F(2) - F(0) = 8/3 - 0
Answer: 8/3 ≈ 2.667

Example 3: Trigonometric

Q: Find ∫cos(x)dx

Solution: Since d/dx[sin(x)] = cos(x), we have: Answer: sin(x) + C

Integration Techniques

1. Substitution (u-substitution)

When integrand contains f(g(x))·g'(x), let u = g(x).

Example: ∫2x·e^(x²)dx

Let u = x², du = 2x dx
∫e^u du = e^u + C = e^(x²) + C

2. Integration by Parts

∫u dv = uv - ∫v du

Example: ∫x·e^x dx

u = x, dv = e^x dx
Result: x·e^x - e^x + C

Applications

1. Area Under Curves

∫[a,b] f(x)dx gives the signed area between f(x) and x-axis from a to b.

2. Volume of Solids

Disk method: V = π∫[a,b] [f(x)]² dx

3. Physics Applications

Displacement: ∫v(t)dt (velocity→position)
Work: W = ∫F(x)dx
Probability: P(a ≤ X ≤ b) = ∫[a,b] f(x)dx

Frequently Asked Questions

What's the difference between ∫ and Σ?

∫ (integral) represents continuous summation over an interval. Σ (sigma) represents discrete summation of individual terms. Integration is the limit of Riemann sums as intervals approach zero.

Why is there +C in indefinite integrals?

The derivative of any constant is zero, so when reversing differentiation, we must account for an unknown constant. For example, both F(x) = x² and F(x) = x² + 5 have derivative 2x.

How do I know which integration technique to use?

Direct integration: Standard forms (x^n, sin, cos, e^x)
Substitution: When you see f(g(x))·g'(x)
Parts: Product of power and exponential/trig
Partial fractions: Rational functions

Can all functions be integrated?

Indefinite: Some functions (like e^(x²)) have no elementary antiderivative, though the integral exists.
Definite: Most continuous functions can be integrated numerically using methods like Simpson's Rule.

What's numerical integration?

When symbolic integration is difficult/impossible, numerical methods approximate ∫[a,b]f(x)dx using:

Trapezoidal Rule: Approximates area with trapezoids
Simpson's Rule: Uses parabolic arcs (more accurate)
Monte Carlo: Random sampling

When is the integral negative?

∫[a,b]f(x)dx is negative when f(x) < 0 on [a,b], meaning the curve is below the x-axis. The integral gives "signed area."