Calculate Integral Rules

Integral calculus is a fundamental tool in mathematics and science. Understanding integral rules allows you to solve complex problems efficiently. This guide explains the key integral rules and provides a calculator to apply them.

Basic Integral Rules

Integral calculus deals with finding the area under a curve. The basic integral rules provide shortcuts for finding integrals of common functions. These rules form the foundation for more advanced techniques.

The integral of a function f(x) with respect to x is written as ∫f(x)dx. The result is called the antiderivative of f(x).

Constant Rule

The integral of a constant is the constant multiplied by the variable of integration.

∫a dx = a·x + C

Power Rule

The power rule applies to functions of the form x^n where n is not equal to -1.

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

Sum and Difference Rules

You can integrate the sum or difference of functions by integrating each term separately.

∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

Constant Multiple Rule

A constant can be factored out of an integral.

∫a·f(x)dx = a·∫f(x)dx

Power Rule

The power rule is one of the most frequently used integral rules. It provides a straightforward method for finding the integral of a function of the form x^n.

Remember that the power rule does not apply when n = -1. For that case, you need to use the natural logarithm function.

Example

Find the integral of x³.

∫x³ dx = (x⁴)/4 + C

Using the power rule, we increase the exponent by 1 and divide by the new exponent. The constant of integration C represents the family of solutions.

Substitution Rule

The substitution rule, also known as u-substitution, is a technique for integrating composite functions. It involves substituting part of the integrand with a new variable.

If f'(x) = g(x), then ∫f(x)·g(x)dx = ∫u du

Steps

Identify u and du
Find the integral in terms of u
Substitute back in terms of x

Example

Find the integral of 2x·e^(x²).

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

Integration by Parts

Integration by parts is a technique for finding the integral of a product of two functions. It is based on the product rule for differentiation.

∫u dv = uv - ∫v du

Steps

Choose u and dv
Find du and v
Apply the integration by parts formula

Example

Find the integral of x·e^x.

Let u = x, dv = e^x dx
Then du = dx, v = e^x
∫x·e^x dx = x·e^x - ∫e^x dx = x·e^x - e^x + C

Worked Examples

Let's apply these rules to solve some common integral problems.

Example 1: Basic Power Rule

Find ∫(3x² + 2x - 5) dx.

∫(3x² + 2x - 5) dx = (3x³)/3 + (2x²)/2 - 5x + C = x³ + x² - 5x + C

Example 2: Substitution Rule

Find ∫x·cos(x²) dx.

Let u = x², du = 2x dx
∫x·cos(x²) dx = (1/2)∫cos(u) du = (1/2)sin(u) + C = (1/2)sin(x²) + C

Example 3: Integration by Parts

Find ∫x·ln(x) dx.

Let u = ln(x), dv = x dx
Then du = (1/x) dx, v = (x²)/2
∫x·ln(x) dx = (x²/2)ln(x) - ∫(x²/2)(1/x) dx = (x²/2)ln(x) - (1/2)∫x dx
= (x²/2)ln(x) - (x²)/4 + C

FAQ

What is the difference between definite and indefinite integrals?

An indefinite integral represents a family of functions (including the constant of integration C), while a definite integral calculates the exact area under a curve between specified limits.

When should I use substitution versus integration by parts?

Use substitution when you have a composite function where one function is the derivative of another. Use integration by parts when you have a product of two functions and one of them can be easily integrated.

What is the constant of integration C?

The constant of integration C represents the infinite number of solutions to a differential equation. It accounts for any initial conditions that might be present in a specific problem.

How do I know which integral rule to apply?

Start by identifying the form of the integrand. If it's a power function, use the power rule. If it's a product of functions, consider substitution or integration by parts. For more complex cases, look for patterns that match known integral formulas.