Solve Indefinite Integral Calculator

An indefinite integral represents a family of functions whose derivatives are equal to the integrand. This calculator helps you find antiderivatives by applying integration rules and techniques.

What is an indefinite integral?

An indefinite integral, also known as an antiderivative, is a function that represents all the functions whose derivative is equal to the integrand. It's written with an integral sign and includes a constant of integration (C).

∫f(x) dx = F(x) + C

where F'(x) = f(x)

The constant C accounts for the infinite number of functions that could have the same derivative. For example, the derivative of both x² and x² + 5 is 2x, so both are antiderivatives of 2x.

Key characteristics of indefinite integrals

Represent a family of functions
Include an arbitrary constant (C)
Differ by at most a constant
Have an infinite number of solutions

Basic rules for integration

Integration follows several fundamental rules that make solving integrals easier. These rules are analogous to the differentiation rules but in reverse.

Power rule

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

Constant multiple rule

∫k·f(x) dx = k·∫f(x) dx

Sum rule

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

Exponential rule

∫eˣ dx = eˣ + C

Natural logarithm rule

∫(1/x) dx = ln|x| + C

Remember that integration is the reverse process of differentiation. The rules for integration are derived from differentiation rules.

Common integrals to remember

Many integrals appear frequently in calculus problems. Memorizing these common integrals can save time and effort when solving problems.

Integrand	Antiderivative
∫xⁿ dx	(xⁿ⁺¹)/(n+1) + C (n ≠ -1)
∫eˣ dx	eˣ + C
∫aˣ dx	(aˣ)/ln(a) + C
∫sin(x) dx	-cos(x) + C
∫cos(x) dx	sin(x) + C
∫sec²(x) dx	tan(x) + C
∫csc²(x) dx	-cot(x) + C
∫sec(x)tan(x) dx	sec(x) + C
∫csc(x)cot(x) dx	-csc(x) + C
∫1/x dx	ln\|x\| + C

These common integrals form the foundation of integral calculus. Practice integrating these functions to build your integration skills.

How to solve indefinite integrals

Solving indefinite integrals involves applying integration rules and techniques. Here's a step-by-step approach:

Identify the integrand and determine if it matches any of the basic integration rules
Apply the appropriate integration rule to find the antiderivative
Add the constant of integration (C) to represent the family of solutions
Simplify the result if possible
Verify your answer by differentiating the result to ensure you get back to the original integrand

Step-by-step example

Let's solve ∫(3x² + 2x - 5) dx step by step:

Break the integral into three parts: ∫3x² dx + ∫2x dx - ∫5 dx
Apply the power rule to each term:
- ∫3x² dx = 3·(x³/3) + C = x³ + C
- ∫2x dx = 2·(x²/2) + C = x² + C
- ∫5 dx = 5x + C
Combine the results: x³ + x² - 5x + C
Verify by differentiating: d/dx (x³ + x² - 5x + C) = 3x² + 2x - 5

Always verify your solutions by differentiation to ensure they're correct.

Worked examples

Here are several examples of solving indefinite integrals using the rules we've learned.

Example 1: Basic polynomial

Find ∫(4x³ - 2x + 7) dx

Solution:

∫4x³ dx = 4·(x⁴/4) + C = x⁴ + C
∫-2x dx = -2·(x²/2) + C = -x² + C
∫7 dx = 7x + C
Combine: x⁴ - x² + 7x + C

Example 2: Exponential function

Find ∫(5eˣ - 3e⁻ˣ) dx

Solution:

∫5eˣ dx = 5eˣ + C
∫-3e⁻ˣ dx = -3·(e⁻ˣ)/-1 + C = 3e⁻ˣ + C
Combine: 5eˣ + 3e⁻ˣ + C

Example 3: Trigonometric function

Find ∫(2sin(x) - 3cos(x)) dx

Solution:

∫2sin(x) dx = -2cos(x) + C
∫-3cos(x) dx = -3sin(x) + C
Combine: -2cos(x) - 3sin(x) + C

FAQ

What is the difference between definite and indefinite integrals?: An indefinite integral represents a family of functions (with a constant of integration), while a definite integral represents a specific numerical value over an interval.
Why do we need the constant of integration in indefinite integrals?: The constant accounts for the infinite number of functions that could have the same derivative. It represents the arbitrary constant in the general solution.
How do I know when to use integration by parts?: Integration by parts is useful when the integrand is a product of two functions. The formula is ∫u dv = uv - ∫v du.
What if I can't find the antiderivative of a function?: If you can't find an antiderivative using basic rules, you may need to use more advanced techniques like substitution, integration by parts, or partial fractions.
How can I check if my antiderivative is correct?: Differentiate your result and see if you get back to the original integrand. This is the best way to verify your solution.