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The concept of integration is important not only in the mathematical field but also in many science and engineering fields. The integration chapter holds maximum weightage in the board exam, and students also find it challenging. To help students get conceptual clarity and study for the exams, we have added important formulas and questions.
To master the chapter, students need to understand several integration methods, application-based problems, and properties in detail. In this writing, we will discuss the basic concept of integration, revise some important formulas, and give a list of important questions to help the students prepare for their exams.
Class 12 Chapter 7 is very important for the students as many questions are framed from this chapter in board examinations. In Important Questions For Class 12 Maths Chapter 7 Integration, topics like integration as an inverse process of differentiation, some properties of indefinite integrals, comparison between differentiation and integration, methods of integration are covered.
PREMIUM EDUCART QUESTIONS
(Most Important Questions of this Chapter from our 📕)
In the table given below, we have provided the links to Class 12 Maths Chapter 7 Important Questions with Solutions PDFs for Integrals. You can download them without having to share any login info.
Integration is nothing but the reverse process of differentiation. Where differentiation helps to find the rate of change of a function, integration finds the total accumulation of quantities. Integration can be used in any field, like finding areas and volumes as well as for problems in physics, economics, and engineering.
Below are some important formulas of integration that the students must remember to solve integration questions:
Power Rule:
∫ xn dx = xn+1 / n+1 + C, n is not equal to -1
Exponential Function:
∫ ex dx = ex + C
Trigonometric Functions:
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ secx dx = tan x + C
∫ cosecx dx = -cot x + C
∫ sec x tan x dx = sec x + C
∫ cosec x cot x dx = -cosec x + C
Logarithmic Functions:
∫ 1/x dx = ln |x| + C
Integration by Substitution:
If u = g(x), then:
∫ f(g(x)) g'(x) dx = ∫ f(u) du
Integration by Parts:
∫ u dv = uv - ∫ v du
Standard Definite Integration:
∫ xn dx = [ xn+1 / n+1 ]
Now let's see some common types of integration problems that can come in Class 12.
Most of the problems involve polynomials or rational functions. The two most common methods of solving these problems are using the power rule or substitution.
Example:
∫ (x3 + 2x2 - 4x + 1) dx
To solve this, use the power rule for each term in the polynomial.
The method is useful if the integrand contains a composite function. It may be possible to convert the function into an easily computable form by using the substitution method.
Example:
∫ (2x) / (x2 +1) dx
Here, we can use the substitution u = x2 + 1 to get an expression that is easier to integrate.
The integration by parts formula is derived from the product rule of differentiation and is very useful when integrating products of functions. To make the problem easy, it is advised to choose u and dv carefully.
Example:
∫ x sin x dx
Here, choose u = x and dv = sin x dx, and then use the integration by parts formula.
The result of definite integration is a numerical value obtained when finding the integral of a function over a given interval [a, b].
Example:
∫ (3x2 - 2x + 1) dx, interval given is [0, 1]
To solve this question, integrate the function first and then apply the limits.
These are problems that contain standard trigonometric functions, often requiring knowledge of formulas or the use of substitution techniques.
Example:
∫ sin2 x dx
This integral can be solved with the help of the identity sin2 x = 1 - cos(2x) / 2, which helps for solving the integral much more easily.
To get a better understanding of the concept of integration, the students should practice the following integration techniques:
This method helps integrate rational functions that have a factorable denominator. The fraction is broken into smaller parts, thus making it easier to integrate each part.
For integrals that include square roots of quadratic expressions, trigonometric substitution can be a very useful tool. The technique makes use of identities like sin2 θ + cos2 θ = 1 to simplify the integral.
These recursive formulas are useful in reducing the power of the integrand. They are commonly applied to integrals that involve powers of trigonometric functions.
These integrals have infinite limits or the integrand is discontinuous. Appropriate techniques for integration have to be used to solve such problems.
This section has the most important questions of the Class 12 Integration.
1. Find the integral of:
∫ 1 / (√1-x2) dx
2. Evaluate the definite integral:
∫ sin x dx, in the interval [0, π/2]
3. Solve the following using integration by parts:
∫ x ex dx
4. Find the area under the curve for:
y = x2 from x = 0 to x = 3
5. Evaluate the integral:
∫ dx / x ln x
6. Evaluate the integral:
∫ sin2 x dx
7. Solve:
∫ 1 / (x2 + 4) dx
8. Find the volume of the solid formed by rotating the curve y = x2 about the x-axis from x = 0 to x = 2.
9. Find the area bounded by the curve y = x2 and the x-axis between x = 0 and x = 2.
10. Find the volume of a solid formed by rotating the curve y = x2 between x = 0 and x = 1 about the x-axis.
Integration is a very important tool in mathematics, especially when solving problems involving areas, volumes, and motion. Mastering integration in Class 12 requires a solid conceptual understanding combined with proficiency in various techniques. By working through a variety of integration problems, students can build the confidence to tackle even the most challenging integrals. The questions and methods discussed above offer a thorough guide for preparing for the Class 12 integration unit, ensuring that students are ready for their exams and future studies.
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