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INTEGRAL
In mathematics, an integral is a mathematical operation that allows for the calculation of
the area or volume of a shape or the length of a curve
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There are two main types of integrals: indefinite integrals and definite integrals
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One common problem involving integrals is finding the area under a curve
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The area under this curve can be found
using the definite integral:
∫y dx = ∫x^2 dx
To solve this problem, we must first find the indefinite integral:
∫x^2 dx = (1/3)x^3 + C
Where C is a constant
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USES ON INTEGRALS

Integrals have a wide range of uses in a variety of fields, including mathematics, physics,
engineering, and other sciences
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Calculating areas and volumes: One of the most common uses of integrals is to calculate
the area or volume of a shape
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2
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For example, the velocity of an
object can be found by integrating its acceleration over time
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Solving differential equations: Differential equations are used to describe how quantities
change over time, and can be solved using integrals
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These are just a few examples of the many uses of integrals in mathematics and the
sciences
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INTEGRAL FORMULAS

Certainly, here are detailed explanations of each of the integral formulas that I listed earlier,
along with examples of how to use them:

1
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For example, suppose we want to evaluate the integral
∫01x^2dx
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Then, the integral is given by:
∫01x^2dx = F(1) - F(0) = (1^3)/3 - (0^3)/3 = 1/3
2
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For example, suppose we want to evaluate the integral ∫0π/2sin(x^2)dx
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Then, the integral becomes:
∫0π/2sin(x^2)dx = ∫0πsin(u)du/2x = ∫0πsin(u)du/2√(u)
3
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For example, suppose we want to
evaluate the integral ∫0π/2cos(x)sin(x)dx
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This gives us:
∫0π/2cos(x)sin(x)dx = [sin(x)cos(x)]π/2 - ∫0π/2du = [sin(x)cos(x)]π/2 - u|π/2 = (1/2)π 1/2 = π/4
4
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For example,
suppose we want to evaluate the integral ∫0π/2sin(x)cos(x)dx
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The Exponential Function Integral Formula: This states that if a is a nonzero constant, then:
∫a^x dx = (a^x)/ln(a) + C

This formula can be used to evaluate integrals involving exponential functions
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We can use the formula ∫a^x dx = (a^x)/ln(a) + C with
a = e^2 to get:
∫10e^(2x)dx = [(e^2)^x]/ln(e^2) + C = (e^2x)/2 + C
6
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For example, suppose
we want to evaluate the integral ∫10ln(2x)dx
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The Inverse Trigonometric Function Integral Formulas: These include:
∫1/(1+x^2)^(1/2) dx = tan^(-1)(x) + C ∫1/(a^2+x^2)^(1/2) dx = a^(-1)tan^(-1)((x)/a) + C
These formulas can be used to evaluate integrals involving inverse trigonometric functions
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We can use the formula
∫1/(1+x^2)^(1/2) dx = tan^(-1)(x) + C to get:
∫01/(1+x^2)^(1/2)dx = [tan^(-1)(x)]1 - [tan^(-1)(0)]0 + C = tan^(-1)(1) - tan^(-1)(0) + C = π/4

8
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For
example, suppose we want to evaluate the integral ∫01sinh(x)dx

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