Integration is a fundamental concept in calculus that refers to the process of finding the integral of a function. The integral represents the accumulation or summation of quantities and is often interpreted as finding the area under a curve or the total accumulated value over a specific interval. In simpler terms, it helps calculate things like the area under a curve, the total distance traveled over time, or the total amount of a quantity (such as mass or volume) over a given range.
While the concept of integration provides a powerful tool for calculating areas, volumes, and accumulated quantities, the process itself can often be complex. To simplify these calculations, mathematicians have developed a set of integration rules. These rules provide a structured approach to solving integrals efficiently, allowing us to handle a wide range of functions and scenarios.
In the following sections, we will explore the most commonly used integration rules that form the foundation of integral calculus. These rules not only simplify the process but also provide strategies for tackling more complex integrals, helping you to integrate with confidence and precision.
Power Rule
The Power Rule is used to integrate functions of the form xn. The rule states:
\bold{\int x^n dx= \frac{x^{n+1}}{n \ + \ 1} + C} (for n ≠ -1)
Where C is the constant of integration. If n = -1, the integral becomes:
∫ x-1 dx = ln |x| + C
Example: Calculate ∫x2 dx:
∫ x2 dx = x3/3 + C
Read more: Power Rule
Constant Rule
The Constant Rule states that the integral of a constant c with respect to x is:
∫ c dx = cx + C
Where, C is the constant of integration.
Example: Calculate ∫ 5 dx:
∫ 5 dx = 5x + C
Sum/Difference Rule
The integral of the sum (or difference) of two functions is the sum (or difference) of their integrals.
∫ (f(x) + g(x)) dx = ∫ f(x) dx + ∫ g(x) dx
∫ (f(x) − g(x)) dx = ∫ f(x) dx − ∫ g(x) dx
Example: Calculate ∫ (x2 + 3x)dx:
∫ (x2 + 3x)dx = ∫ x2dx + ∫3x dx = \frac{x^3}{3} + \frac{3x^2}{2}+C
Constant Multiple Rule
The integral of a constant multiplied by a function is the constant multiplied by the integral of the function:
∫ c ⋅ f(x) dx = c ⋅ ∫f(x) dx
Where c is a constant.
Example: Calculate ∫5x2 dx
∫ 5x2dx = 5 ⋅ ∫x2dx = 5 \cdot\frac{x^3}{3} + C = \frac{5x^3}{3} + C
Read More: Constant Multiple Rule
Substitution Rule
For the substitution u = g(x), the integral becomes easier by replacing the original variable with u.
∫ f(g(x)) ⋅ g′(x)dx = ∫ f(u)du
Example: Calculate ∫2x ⋅ cos(x2)dx
Let u = x2, so du = 2x dx. The integral becomes:
∫ cos(u) du = sin(u) + C = sin(x2) + C
Read More: Integration by Substitution
ILATE Rule
The ILATE Rule is a helpful guideline for choosing the best function to differentiate when using integration by parts. It stands for:
ILATE Rule for IntegrationThe rule suggests that when you're solving an integral using integration by parts, you should pick the function from this list in the order of ILATE. This helps simplify the integral, as differentiating these functions in this order makes the problem easier to solve.
Example: Calculate ∫ x ⋅ ln x dx
According to ILATE, the logarithmic function (ln x) is higher in the hierarchy than the algebraic function (x), so we choose:
Apply integration by parts formula:
∫u dv = uv − ∫v du
If
- u = ln x, so du = 1/x dx
- dv = x dx, so v = ∫x dx = x2/2
Substitute into the formula
Substitute u, v, du, and dv into the formula: ∫x ⋅ ln x dx = uv − ∫v du
= \ln x \cdot\frac{x^2}{2} - \int\frac{x^2}{2} \cdot \frac{1}{x}dx
Simplify the remainig integral:
\int x \cdot \ln x \ dx = \frac{x^2}{2} \ln x - \frac{x^2}{4}+ C
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