Integrated Formula, All Trigonometric Functions, Classification
Integrated Formula: Learn key integration formulas for Class 12, including trigonometric, logarithmic, and power rules, to simplify solving integrals in calculus.
Shruti Dutta16 Dec, 2024
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Integrated Formula: Integration is a fundamental concept in calculus that involves finding the antiderivative of a given function. It is the reverse process of differentiation and plays a crucial role in various branches of mathematics, science, and engineering.
In simple terms, an Integrated Formula helps in determining the accumulated quantity, such as the area under curves, the total distance travelled, or the volume of solids. Mastering these integrated formula is essential for Class 12 students, as they form the foundation for more advanced topics in calculus and its applications. These formulas allow students to efficiently solve integrals without the need for lengthy manual calculations, making integration a powerful tool in both academic and real-world contexts.Also Check: Pie Chart
What are Integrated Formula?
Integrated formula are mathematical expressions or rules used to compute the integral of a function. An integral is essentially the accumulation or summation of quantities, and in calculus, it is used to find areas under curves, volumes of solids, and other quantities that can be expressed as a sum. Integrated formula provide specific techniques and shortcuts to calculate these integrals without having to rely on basic principles or numerical methods every time. There are different types of integrated formula, each designed to handle specific kinds of functions. These formulas are critical in various fields, such as physics, engineering, economics, and probability theory, where integrals are frequently encountered. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Copy-of-Corurious-Jr-Reel-2-Landscape-1-2-1.mp4"][/video]Also Check: Natural Numbers
Classification of Integrated Formula
Integrated formula are a fundamental aspect of calculus and mathematical analysis. They provide a systematic way to evaluate integrals, which are critical for solving problems in physics, engineering, economics, and many other fields. The classification of integrated formula is based on the types of functions involved in the integrals. Below is an explanation of the different categories under which integrated formula can be classified:1. Rational Functions
Rational functions are the ratio of two polynomials. The general form is π(π₯)π(π₯), where both π(π₯)and π(π₯) are polynomials. Integration of rational functions typically involves methods such as partial fraction decomposition.Example:
β«1π₯2+1ππ₯β«
2. Irrational Functions
Irrational functions involve roots of polynomials or other functions. These are typically expressions that contain square roots, cube roots, or higher-order roots.Example:
β«π₯2+1ππ₯β«
The integration of irrational functions often requires substitution or trigonometric identities.3. Hyperbolic Functions
Hyperbolic functions are analogues of trigonometric functions but for hyperbolas. They are defined in terms of exponential functions and have integrals similar to their trigonometric counterparts. Common hyperbolic functions include:sinh(π₯),cosh(π₯),tanh(π₯),sech(π₯),coth(π₯)
Example:
β«cosh(π₯)ππ₯
The integral of cosh(π₯)sinh(π₯)+πΆ [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Curious-Jr-Ad.mp4"][/video]4. Inverse Hyperbolic Functions
Inverse hyperbolic functions are the inverses of the hyperbolic functions. These include arsinh(π₯),arcosh(π₯),artanh(π₯) and others.Example:
β«1+π₯2ππ₯
The result is arsinh(π₯)+πΆ5. Trigonometric Functions
Trigonometric functions like sine, cosine, tangent, and others have well-known integrals. These functions are frequently encountered in applications involving periodic motion, waves, and oscillations. Common trigonometric functions include: sin(π₯),cos(π₯),tan(π₯),sec(π₯),cot(π₯),csc(x).Example:
β«sin(π₯)ππ₯
The result is βcos(π₯)+πΆ
6. Inverse Trigonometric Functions
Inverse trigonometric functions are the inverses of the standard trigonometric functions. These functions include arcsin(π₯),arccos(π₯),arctan(π₯) etc. Their integrals often arise in problems involving angles or circular motion.Example:
β«1βπ₯2ππ₯
The result is arcsin(π₯)+πΆarcsin(x)+C.7. Exponential Functions
Exponential functions involve the constant π raised to the power of a variable. The integral of an exponential function is one of the most straightforward integrations. Common exponential functions include:ππ₯,πππ₯,ππ₯2e
Example:
β«ππ₯ππ₯β«e
The result isππ₯+πΆ
8. Logarithmic Functions
Logarithmic functions involve the natural logarithm ln(π₯)or logarithms with different bases. These functions have an integral that is frequently encountered in problems involving rates of change, growth, or decay.Example:
β«1π₯ππ₯β«
lnβ£xβ£+C.
Also Check: Pie Chart
Integration by Parts (Product Rule of Integration)
The integration by parts formula is a powerful technique used in calculus to integrate the product of two functions. It is based on the product rule of differentiation and is commonly referred to as the product rule of integration. The integration formula is particularly useful when we have an integral of the form β«π’ππ£, where π’ and π£ are two differentiable functions of π₯. The formula for integration by parts is expressed as:β«udv=uvββ«vdu
Where :- U is the first function,
- dv is the second function (in differential form),
- du is the derivative of u, and
- v is the integral of dv.
β«udv=uβ«vdxββ«(u'β«vdx)dx
Here, π’'is the derivative of π’ and β«vdx represents the antiderivative of π£ Choosing the First Function (u): To apply the integration by parts method effectively, we must decide which function should be assigned to π’ and which to π£ A useful guideline for choosing π’ is the ILATE or LIATE rule, which suggests a priority order for different types of functions:- L : Logarithmic functions (e.g., ln(x))
- I Inverse trigonometric functions (e.g.,a rctan(x))
- A : algebraic functions (e.g., polynomials like π₯2 )
- T : trigonometric functions (e.g., sin(π₯), cos(π₯))
- E : Exponential functions (e.g., ππ₯ )
Also Check: CM to Feet
Integration Formulas Class 12
In Class 12 Mathematics, integration is an essential topic that deals with finding the integral or antiderivative of functions. These formulas are used to solve a wide range of problems, from basic polynomials to more complex functions like trigonometric, logarithmic, and exponential expressions. Knowing these formulas and their applications is crucial for students to understand the concepts of integration and to solve real-life problems effectively. Below is a list of key integration formulas that are commonly covered in Class 12 Mathematics.| Rule/Formula | Expression |
| Constant rule | β«c dx = cx + C, where c is a constant. |
| Power rule | β«x^n dx = (x^(n+1))/(n+1) + C, where n β -1. |
| Sum and difference rules | β«(f(x) Β± g(x)) dx = β«f(x) dx Β± β«g(x) dx. |
| Constant multiple rule | β«c * f(x) dx = c * β«f(x) dx. |
| Trigonometric functions | β«sin(x) dx = -cos(x) + C, β«cos(x) dx = sin(x) + C, β«sec^2(x) dx = tan(x) + C, β«sec(x)tan(x) dx = sec(x) + C |
| Inverse trigonometric functions | β«csc^2(x) dx = -csc(x)cot(x) + C, β«sec(x)cot(x) dx = ln|sec(x) + tan(x)| + C. |
| Logarithmic functions | β«(1/x) dx = ln|x| + C, β«ln(x) dx = xln(x) β x + C. |
| Exponential functions | β«e^x dx = e^x + C. |
| Integration by substitution | β«f(g(x))g'(x) dx = β«f(u) du, where u = g(x). |
| Integration by parts | β«u dv = uv β β«v du, where u and v are functions of x. |
| Trigonometric substitution | β«(a^2 β x^2)^(1/2) dx = a β«sec^2(ΞΈ) dΞΈ, where x = a sin(ΞΈ) or x = a tan(ΞΈ) |
| partial fraction decomposition | β«(f(x)/(ax+b)) dx = ln|ax+b| + C |
Also Check: Even Numbers
Integrated Formula of Trigonometric Functions
The integrated formula for trigonometric functions provides simple and efficient ways to calculate areas under curves involving sine, cosine, secant, and other trigonometric expressions. By mastering these formulas, students can easily tackle integration problems in Class 12 mathematics. Below are the key integration formulas for trigonometric functions that are essential for solving such problems. β« cos x dx = sin x + C β« sin x dx = -cos x + C β« sec2x dx = tan x + C β« cosec2x dx = -cot x + C β« sec x tan x dx = sec x + C β« cosec x cot x dx = -cosec x + C β« tan x dx = log |sec x| + C β« cot x dx = log |sin x| + C β« sec x dx = log |sec x + tan x| + C β« cosec x dx = log |cosec x β cot x| + C| Related Articles | |
| Subtraction Sums | Word Problems |
| What is an Abacus? | Math Puzzles |
| Multiplication Worksheets | Multiplication Sums |
Integrated Formula FAQs
What are integration formulas?
Integration formulas are mathematical expressions that provide specific rules for calculating the integral or antiderivative of a function. These formulas simplify the process of finding integrals for different types of functions, such as polynomials, trigonometric functions, logarithmic functions, and more.
Why are integration formulas important?
Integration formulas are crucial because they allow us to quickly and efficiently compute integrals, which are used to find areas under curves, volumes of solids, and other important quantities in physics, engineering, economics, and other fields.
What is the role of constant CCC in integration formulas?
The constant C represents the "constant of integration." When we find an indefinite integral, we are essentially reversing the process of differentiation. Since differentiation removes constant terms, the constant c accounts for all possible constant values that could have been present in the original function.
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