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Integration

Integration

The process of finding an integral, either a definite integral or an indefinite integral.

 

 

See also

Integration methods

Key Formula

∫f(x) dx=F(x)+CwhereF′(x)=f(x)\int f(x)\,dx = F(x) + C \quad \text{where} \quad F'(x) = f(x)
Where:
  • f(x)f(x) = The function being integrated (called the integrand)
  • F(x)F(x) = The antiderivative — a function whose derivative equals f(x)
  • CC = The constant of integration, representing any constant value
  • dxdx = Indicates integration is performed with respect to x

Worked Example

Problem: Find the indefinite integral of f(x) = 3x².
Step 1: Identify the integrand and apply the power rule for integration. The power rule states that the integral of xⁿ is xⁿ⁺¹ divided by (n + 1).
∫xn dx=xn+1n+1+C(n≠−1)\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)
Step 2: Pull the constant 3 out of the integral and apply the power rule with n = 2.
∫3x2 dx=3⋅x2+12+1+C=3⋅x33+C\int 3x^2\,dx = 3 \cdot \frac{x^{2+1}}{2+1} + C = 3 \cdot \frac{x^3}{3} + C
Step 3: Simplify the expression.
=x3+C= x^3 + C
Step 4: Verify by differentiating: the derivative of x³ + C is 3x², which matches the original integrand.
ddx(x3+C)=3x2✓\frac{d}{dx}(x^3 + C) = 3x^2 \checkmark
Answer: The integral of 3x² is x³ + C.

Another Example

Problem: Evaluate the definite integral of 2x from x = 1 to x = 4.
Step 1: Find the antiderivative of 2x using the power rule.
∫2x dx=x2+C\int 2x\,dx = x^2 + C
Step 2: Apply the Fundamental Theorem of Calculus: evaluate the antiderivative at the upper limit and subtract its value at the lower limit. The constant C cancels out.
∫142x dx=[x2]14=42−12\int_1^4 2x\,dx = \left[x^2\right]_1^4 = 4^2 - 1^2
Step 3: Compute the result.
=16−1=15= 16 - 1 = 15
Answer: The definite integral equals 15, which represents the net area under the curve y = 2x between x = 1 and x = 4.

Frequently Asked Questions

What is the difference between integration and differentiation?
Differentiation finds the rate of change of a function (its derivative), while integration reverses that process to recover the original function (its antiderivative). The Fundamental Theorem of Calculus formally connects the two: integrating a derivative returns the original function, and differentiating an integral returns the original integrand.
Why do we add '+ C' when integrating?
The constant of integration C accounts for the fact that many different functions share the same derivative. For example, x², x² + 5, and x² − 3 all have derivative 2x. Since integration reverses differentiation, you must include + C to represent all possible antiderivatives. In definite integrals, C cancels out and is not needed.

Indefinite Integration vs. Definite Integration

Indefinite integration produces a family of functions (an antiderivative plus a constant C), such as ∫2x dx = x² + C. Definite integration evaluates the integral between two specific bounds and produces a single number, such as ∫₀³ 2x dx = 9. The indefinite integral answers 'what function has this derivative?' while the definite integral answers 'what is the total accumulated value over this interval?'

Why It Matters

Integration is essential across science, engineering, and economics. Physicists use it to compute work, energy, and displacement from velocity. Engineers rely on it to find areas, volumes of solids, and the total charge in a circuit. Economists use integration to calculate consumer surplus and total cost from marginal cost functions.

Common Mistakes

Mistake: Forgetting the constant of integration (+C) in indefinite integrals.
Correction: Every indefinite integral must include + C because infinitely many functions share the same derivative. Only definite integrals (with specific bounds) produce a single number without C.
Mistake: Applying the power rule incorrectly by forgetting to increase the exponent by 1 before dividing.
Correction: The power rule requires you to first add 1 to the exponent and then divide by the new exponent: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. A common error is writing xⁿ/n instead.

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