Indefinite Integral

Indefinite Integral

The family of functions that have a given function as a common derivative. The indefinite integral of f(x) is written
∫ f(x) dx.

Example: integral of x² dx = (1/3)x³ + C

Key Formula

\int f(x)\,dx = F(x) + C

Where:

$f(x)$ = The integrand — the function you are integrating
$F(x)$ = Any particular antiderivative of f(x), meaning F'(x) = f(x)
$C$ = The constant of integration, representing any real number
$dx$ = Indicates that the variable of integration is x

Worked Example

Problem: Find the indefinite integral of f(x) = 3x².

Step 1: Write the integral in standard notation.

\int 3x^2\,dx

Step 2: Apply the power rule for integration: raise the exponent by 1, then divide by the new exponent.

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Step 3: Substitute n = 2 and include the coefficient 3.

3 \cdot \frac{x^{2+1}}{2+1} + C = 3 \cdot \frac{x^3}{3} + C

Step 4: Simplify the expression.

x^3 + C

Step 5: Verify by differentiating: the derivative of x³ + C is 3x², which matches f(x). ✓

\frac{d}{dx}(x^3 + C) = 3x^2

Answer: ∫ 3x² dx = x³ + C

Another Example

This example involves a trigonometric function combined with a constant, showing how the sum rule lets you integrate term by term. It also demonstrates that you only need one constant C at the end, not one per term.

Problem: Find the indefinite integral of f(x) = 4cos(x) + 5.

Step 1: Write the integral and split it using the sum rule.

\int (4\cos x + 5)\,dx = \int 4\cos x\,dx + \int 5\,dx

Step 2: Integrate the first part. The antiderivative of cos(x) is sin(x).

\int 4\cos x\,dx = 4\sin x

Step 3: Integrate the second part. The antiderivative of a constant k is kx.

\int 5\,dx = 5x

Step 4: Combine the results and add the constant of integration.

4\sin x + 5x + C

Answer: ∫ (4cos(x) + 5) dx = 4sin(x) + 5x + C

Frequently Asked Questions

What is the difference between an indefinite integral and a definite integral?

An indefinite integral produces a family of functions (an antiderivative plus C), while a definite integral evaluates the net area under a curve between two specific bounds and produces a single number. The definite integral uses the Fundamental Theorem of Calculus to compute F(b) − F(a), so the constant C cancels out.

Why do you add + C to an indefinite integral?

Because many different functions can share the same derivative. For example, x², x² + 7, and x² − 3 all have the derivative 2x. The constant C accounts for every possible vertical shift. Without it, you would be giving only one specific antiderivative instead of the complete family.

How do you check if an indefinite integral is correct?

Differentiate your answer. If the derivative of your result equals the original integrand f(x), your integral is correct. This works because integration and differentiation are inverse operations. For instance, if you found ∫2x dx = x² + C, differentiating x² + C gives 2x, confirming the answer.

Indefinite Integral vs. Definite Integral

	Indefinite Integral	Definite Integral
Definition	The family of all antiderivatives of a function	The net signed area under a curve between two bounds
Notation	∫ f(x) dx	∫ from a to b of f(x) dx
Result	A function plus a constant: F(x) + C	A single number: F(b) − F(a)
Constant of integration	Required (+ C)	Not needed (cancels out)
Primary use	Finding antiderivatives and general solutions	Computing areas, total change, and accumulated quantities

Why It Matters

Indefinite integrals are foundational to calculus and appear throughout physics, engineering, and economics. When you solve a differential equation — such as finding velocity from acceleration — you are computing an indefinite integral. Mastering indefinite integration is also essential for evaluating definite integrals, since the Fundamental Theorem of Calculus requires you to first find an antiderivative.

Common Mistakes

Mistake: Forgetting the constant of integration C.

Correction: Every indefinite integral must include + C. Without it, your answer represents only one antiderivative rather than the entire family. On exams, omitting C is typically marked as an error.

Mistake: Misapplying the power rule when n = −1.

Correction: The power rule ∫xⁿ dx = xⁿ⁺¹/(n+1) + C does not work when n = −1, because division by zero would occur. Instead, ∫x⁻¹ dx = ∫(1/x) dx = ln|x| + C.

Related Terms

Derivative — The inverse operation of indefinite integration
Function — The integrand and result are both functions
Integral of a Function — General concept encompassing definite and indefinite integrals
Integration Methods — Techniques like substitution and parts for finding antiderivatives
Integral Rules — Power rule, sum rule, and other rules used in integration
Integral Table — Reference list of common indefinite integrals