Unveiling the Art of Finding the Reverse Formula
Comprehensive Definition, Description, Examples & Rules
Introduction
The antiderivative is a function that you can describe as the exact opposite of the function of a derivative. It is similar to the original derivative of the function. The antiderivative function is similar to an indefinite integral. You use the antiderivative to find the total of specific things and also can calculate the amount of distance and time. It is the differentiable function whose derivative is equal to the original function.
Importance in Calculus
The antiderivative also has a significant role to play in calculus. It includes:
- It effectively calculates the total things that help maintain Calculus’s fundamental theorem.
- It also helps to compute integrals of one of the two fundamental operations you use in calculus.
- You can also use the antiderivative to calculate the derivatives of the original function of the calculus.
- You can find the definite integral of a function by using the antiderivatives with the integral calculus.
The formula and theorem of Calculus have a big support from the derivative and the antiderivative calculations.
Understanding Antiderivatives
You can define an antiderivative as a function that is the reverse action of the derivative. Any function the derivative has is just opposite to that of an antiderivative. A particular function can have one or more antiderivatives, and all of these functions form an arbitrary constant. These are a primary part of the definite integrals.
Usage:
- Calculate how long it will take for a certain thing to go in a specific period.
- You can use them to find the total of specific things.
The derivative is a valuable tool in calculus for determining the original form of a function. Finding the derivative is called differentiation, while the reverse process is called antidifferentiation. It’s an effective method for analysing functions in calculus.
Significance of Antiderivatives in Mathematics:
- It helps you calculate and simplify a difficult mathematical calculation using the fundamental theory of Calculus.
- These are related to the definite integrals, where you can follow Calculus’s second fundamental theory and find the functions’ values.
Significance of Antiderivatives in Real-Life Applications:
- In real-life applications, you can use antiderivatives to find a stock’s future value in the stock market.
- Calculating the speed of the car is also possible through antiderivatives.
Antiderivative Formula
The antiderivative formula is divided into different rules in calculus, where you need to follow the basic rules and find the antiderivatives of different combinations of different functions. The most commonly used anti-derivative formula is the power rule in which you follow the formula of:
- ∫xn dx = xn+1/(n + 1) + C
In this calculation, the value of n cannot be equal to -1.
Example:
Find Antiderivative of X³:
The function value of F(x) can be obtained with the indefinite integral of the derivative f(x).
By the Power Rule, the integral x3 concerning x is 14×4 1 4 x 4. The answer is the antiderivative of the function f(x)=x 3 f ( x ) = x 3.
The notations used in the formula are the value of n, and you use the power rule to calculate this antiderivative sum.
Common Antiderivative Rules
To calculate the antiderivative, there are specific antiderivative rules that you have to effectively follow, which will help you to calculate the sums. The rules are:
Power Rule
The power rule uses the power function in it, and the formula you use is:
- ∫xn dx = xn+1/(n + 1) + C
Where n is not equal to -1.
Example:
Find antiderivative of 3x²
f(x) = 3x²
F(x)=∫f(x)dx
Set up the integral to solve.
F(x)=∫3x²dx
Since 33 is constant concerning x, move 33 out of the integral.
3∫x²dx
By the Power Rule, the integral of x² for x is ⅓ x³
3 ( ⅓ X³ + C)
Answer = X³ + C
Exponential Rule
The exponential Rule is when you use the exponent to calculate the antiderivative. The formula is:
- F(x) = 1 c ec⋅c + D, for any constant D, is an antiderivative of f(x).
The theorem does not work for C = 0, and it will be constant.
Example:
∫[4 + x2] dx = ∫4 dx + ∫x2 dx
Trigonometric Rules
Through the trigonometric rules, use specific trigonometric formulas that are available.
These are:
- ∫sin-1x dx = x sin-1x + √(1 – x2) + C.
- ∫cot-1x dx = x cot-1x + (1/2) ln(1 + x2) + C.
- ∫tan-1x dx = x tan-1x – (1/2) ln(1 + x2) + C.
- ∫csc-1x dx = x csc-1x + ln |x + √(x2 – 1)| + C.
- ∫sec-1x dx = x sec-1x – ln |x + √(x2 – 1)| + C.
- ∫cos-1x dx = x cos-1x – √(1 – x2) + C.
Example:
Calculate the Antiderivative of Sinx
By using the direct formula, the antiderivative will be:
–cosx + c
Logarithmic Rule
Follows the natural logarithm function and uses the constant of integration. The formula is:
- ∫ ln x dx = x ln x – x + C,
Where C is the integration constant.
Example:
Antiderivative of logx
Using the formula, the value will be:
(X * Inx – X) / In(10) + c
Techniques for Finding Antiderivatives
There are a few integration techniques that you have to follow while calculating the antiderivatives of a function. These are:
Substitution
The substitution technique is how you can find the antiderivative of an integration using the substitution method. It is the reverse rule that you use to solve an equation of a variable and then substitute the expression into other functions so that you can find the value of a missing function.
Example:
Find the integration of ∫ etan -1x / 1 + X²
Let t = tan-1x …… (1)
dt = (1/ 1+x2 ). dx
I = ∫ et . dt
= et + C …….(2)
Substituting the value of (1) in (2), we have I = etan-1x + C. This is the required integration.
Integration by Parts
The technique of integration by parts is the way you find the derivative of an integral with a particular formula, and you can use it with different parts.
Example:
What is ∫x cos(x) dx?
We have x multiplied by cos(x), so we can use integration by parts.
Select u and v:
- u = x
- v = cos(x)
So now it is in the format ∫u v dx, we can proceed:
Differentiate u: u’ = x’ = 1
Integrate v: ∫v dx = ∫cos(x) dx = sin(x)
x sin(x) − ∫sin(x) dx
x sin(x) + cos(x) + C
Partial Functions
The partial function is when you use a subset of set x or y to calculate the value of the function, and you use a domain of definition or a natural domain to calculate the antiderivative and the integrals.
Example:
Find the integration of ∫ (x² + 1 / x² – 5x + 6) dx
x² + 1 / x² – 5x + 6 = 1 + 5x – 5 / x² – 5x + 6
x² – 5x + 6 = (x – 2) (x – 3)
x² + 1 / x² – 5x + 6 = 1 + 5x – 5 /(x – 2) (x – 3)
Application of Antiderivatives
The antiderivative has a role to play in human life, and the fields you can use it are:
- Physics: It has a great role in physical science, and various physicians used the antiderivative while calculating, and it has also been a part of various case studies of Newton.
- Economics: The antiderivatives also have a big role in calculating economic calculations, and it helps make it very easy. There have been case studies by the Cambridge universities to calculate and mix up the antiderivatives and economics.
- Engineering: The most prominent area where you use the antiderivatives is the engineering sector, which civil engineers and builders use it to calculate the area between a particular shape, which effectively helps the construction process.
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Key Takeaways
- You can use integrals and antiderivatives to calculate particular data in many day-to-day areas.
- The formulas can confuse you, so you must remember the exact challenges and learn the formulas properly.
- There is an essential part of the differential equation in the integral calculation, which you must also understand.
Quiz
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Frequently Asked Questions
The power rule is where you use the power function and the differentiation process of integration to calculate the antiderivative of a particular calculus. The value of the constant in the power role can never be equal to -1.
The substitution method is when you use the integration technique to find the exact value of the function, and it is the reverse rule where you solve an equation of a variable and substitute the expression into other functions where you can find the value of the missing function and calculate the integration.
The trigonometric formulas will be used for the antiderivative of these common functions, and you need to find the exact reverse of the derivatives of these trigonometric formulas. For example:
Antiderivative of Sinx is -Cosx + C.
The integration by parts is the technique you use to find the derivative of an integral and use different parts to calculate the formula and find the exact amount you want. You can use this process to find the integration of the product of two or more functions where you cannot form the integration by using the normal techniques.
