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Title: Algebra Integrals Integration
Description: These notes are typically study materials or lecture notes that cover the topic of integration within the broader field of algebra and calculus. These notes can be used by students, especially in mathematics and science courses, to understand and apply concepts related to integration.
Description: These notes are typically study materials or lecture notes that cover the topic of integration within the broader field of algebra and calculus. These notes can be used by students, especially in mathematics and science courses, to understand and apply concepts related to integration.
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Indefinite Integral - Basic Integration Rules, Problems,
Formulas, Trig Functions, Calculus
The Organic Chemistry Tutor
Indefinite Integral Problems
In this tutorial, we will discuss how to solve indefinite integral problems
using various techniques
...
So, the antiderivative of 4 dx is 4x + c
...
Antiderivative of x Raised to a Constant
When we have to integrate a variable raised to a constant like x raised
to the n, we add one to the exponent and divide the result by that
number
...
Integrating Polynomial Functions
For polynomial functions like x squared minus five x plus six, we
integrate each term separately and add a variable to the constant term
...
We then multiply the result by the reciprocal of the exponent
...
We then move
the x variable back to the denominator and simplify the result
...
The antiderivative of 8x cubed is 2x to the fourth plus c
...
The antiderivative of 5x to the sixth power is (5/7)x to the seventh
power plus c
...
The antiderivative of 3x minus 1 squared dx is 3x cubed over 3 3x squared + x + c, which simplifies to x cubed - x squared + x +
c
...
The antiderivative of 2x plus 1 times x minus 2 dx is 2x cubed over
3 - 3x squared over 2 - 2x + c
...
The antiderivative of 1 over x squared dx is -1 over x + c
...
The antiderivative of 5 divided by x to the fourth is (-5/3)x to the
negative three plus c
...
Here are some examples:
The antiderivative of 1/x is ln x
The antiderivative of e^(4x) is (1/4)e^(4x)
The antiderivative of cos x is sin x and the antiderivative of sin
x is -cos x
The antiderivative of sec^2 x is tan x
If the function is not in a simple form, u substitution can be used
...
For
example:
The antiderivative of x^2sin(x^3) can be found by making u =
x^3 and solving for dx, then replacing x^3 with u
...
How do you find the antiderivative of a constant?
To find the antiderivative of a constant like 4, you just add an x and a
constant value (c) to it
...
What is the antiderivative of e?
The antiderivative of e dz is e times z plus c, since e is a constant
...
For example, the antiderivative of x squared dx is x to the third
divided by three plus c
...
How do you find the antiderivative of square root functions?
To find the antiderivative of square root functions like the square root
of x, we rewrite it as x to the one-half and then add one to the
exponent
...
How do you find fractional antiderivatives?
For fractional antiderivatives like 1 over x squared dx, we rewrite it as x
to the negative two and then add one to the exponent
...
What is an antiderivative?
An antiderivative is a function whose derivative is the original function
...
What is the antiderivative of e^(4x)?
The antiderivative of e^(4x) is (1/4)e^(4x)
...
What is the antiderivative of sin x?
The antiderivative of sin x is -cos x
...
What is u substitution?
U substitution is a method used to find antiderivatives for functions that
are not in a simple form
...
How do you use u substitution to find an antiderivative?
To use u substitution to find an antiderivative, you make a substitution
for a part of the function that simplifies it, then replace the substituted
term with a u variable
...
What is the antiderivative of x^2sin(x^3)?
The antiderivative of x^2sin(x^3) is -(1/3)cos(x^3) + C
Bullet Points
What is an indefinite integral? - An indefinite integral is the
antiderivative of a given function
...
So, the antiderivative of 4 dx is 4x + c
...
How do you find the antiderivative of a variable raised to a
constant? - When we have to integrate a variable raised to a
constant like x raised to the n, we add one to the exponent and
divide the result by that number
...
How do you integrate polynomial functions? - For polynomial
functions like x squared minus five x plus six, we integrate each
term separately and add a variable to the constant term
...
We then multiply the result by the
reciprocal of the exponent
...
We then move
the x variable back to the denominator and simplify the result
...
The antiderivative of 1/x is ln x
The antiderivative of e^(4x) is (1/4)e^(4x)
The antiderivative of cos x is sin x and the antiderivative of sin
x is -cos x
The antiderivative of sec^2 x is tan x
u substitution can be used if the function is not in a simple
form
...
The antiderivative of x^2sin(x^3) can be found by making u =
x^3 and solving for dx, then replacing x^3 with u
Title: Algebra Integrals Integration
Description: These notes are typically study materials or lecture notes that cover the topic of integration within the broader field of algebra and calculus. These notes can be used by students, especially in mathematics and science courses, to understand and apply concepts related to integration.
Description: These notes are typically study materials or lecture notes that cover the topic of integration within the broader field of algebra and calculus. These notes can be used by students, especially in mathematics and science courses, to understand and apply concepts related to integration.