NCERT Solutions For Class 12 Mathematics Chapter 7 Integrals

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NCERT Solutions for Class 12 Mathematics Chapter 7 Integrals are provided in the article below. Integrals can be defined as the representation of the area of a region under a curve. By drawing rectangles, we approximate the actual value of an integral. A definite integral of a function can be represented as the area of the region bounded by its graph of the given function between two points in the line.

Chapter 7 Integrals will carry a total weightage of 6-18 marks in the CBSE Class 12 examination. Around 3-4 long answer questions can come from Double Integral, Continuous Integration, Properties of Definite Integral, Line Integral, Integrals of Particular Function.

Download PDF: NCERT Solutions for Class 12 Mathematics Chapter 7

Class 12 Maths NCERT Solutions Chapter 7 Integrals

NCERT Solutions Class 12 Mathematics Chapter 7 Important Topics

Chapter 7 Integrals is an important topic in the board examination as per CBSE Class 12 exam pattern. In NCERT Class 12 Mathematics Chapter 7, integration as the inverse process of differentiation. integration of a variety of functions by substitution, by partial fractions, and by parts, evaluation of simple integrals of the following types, Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation of definite integrals.

Double Integral

Double integrals are used to find the surface area of a 2-dimensional figure like a circle, square, triangle, pentagon, rectangle and quadrilateral. Mathematically, the double integral is represented as ‘∫∫’ and simple integration forms the basis of doing double integrals. By double integration, we can easily find the area of a rectangular region.

Double integral

Double Integral

The properties of double integrals are as follows:

∫_x=a^b ∫_y=c^d f(x,y)dy.dx = ∫_y=c^d∫_x=a^b f(x,y)dx.dy
∫∫(f(x,y) ± g(x,y)) dA = ∫∫f(x,y)dA ± ∫∫g(x,y)dA
If f(x,y) < g(x,y), then ∫∫f(x,y)dA < ∫∫g(x,y)dA
k ∫∫f(x,y).dA = ∫∫k.f(x,y).dA
∫∫_R∪Sf(x,y).dA = ∫∫_Rf(x,y).dA+∫∫_sf(x,y).dA

Continuous Integration

Continuous integration, in maths, is basically the designating actual numbers to some functions having some potential for negligible data or value for it. The continuous integral value is generally used for expressing displacement, volume, area, and other dimensions of mathematics.

Continuous Integration

Continuous Integration

Types of Continuous Integration

Numerical Integration- This integration is one of the numerical approaches for the evaluation and computation, particularly with computer operations, definite integrals, and generally, solutions of differential equations.

Order of integration- It explains the number of times a time scale decreases for the sole objective of getting it fixed.

Indefinite integration- It is the method of computing indefinite integrals which are basically called anti-derivatives, in calculus.

Properties of Definite Integral

Definite integral is the integral that has upper and lower limits. Integral is the opposite or inverse of differentiation, which is also termed as anti-derivative. A definite integral is a difference between the values of the integral at the specified upper and lower limit of the independent variable. It is represented as; ∫_a^b f(x) dx.

There are many properties of definite integral, those are:

Property 1- _p∫^q f(a) da = _p∫^q f(t) dt

Property 2- _p∫^q f(a) d(a) = – _q∫^p f(a) d(a), Also _p∫^p f(a) d(a) = 0

Property 3- _p∫^qf(a) d(a) = _p∫^r f(a) d(a) + _r∫^q f(a) d(a)

Property 4- _p∫^q f(a) d(a) = _p∫^q f( p + q – a) d(a)

Property 5- _o∫^p f(a) d(a) = _o∫^p f(p – a) d(a)

Property 6- ∫₀^2p f(a)da = ∫₀^pf(a)da +∫₀^pf(2p-a)da…if f(2p-a) = f(a)

Property 7- 2 parts

∫₀²f(a)da = 2∫₀^a f(a) da … if f(2p-a) = f(a)
∫₀²p f(a)da = 0 … if f(2p-a) = -f(a)

Property 8- 2 parts

∫_-p^pf(a)da = 2∫₀^p f(a) da … if f(-a) = f(a) or it’s an even function
∫_-p^pf(a)da = 0 … if f(2p-a) = -f(a) or it’s an odd function

Line Integral

A line integral, in calculus, is an integral where the function to be integrated is evaluated along a curve. A route integral, curvilinear integral, or curve integral are other names for a line integral. Line integrals can be applied in electromagnetic or can be used to estimate the work done on a charged particle traveling along some curve in a force field defined by a vector field.

A line integral along with a smooth curve, C ⊂ U for a scalar field with function f: U ⊆ Rⁿ → R is defined as:

∫_C f(r) ds =

∫_a^b

f[r(t)] |r’(t)| dt

Here, r: [a, b]→C is an arbitrary bijective parametrization of the curve.

r (a) and r(b) gives the endpoints of C and a < b.

A line integral along with a smooth curve C ⊂ U for a vector field with function, F: U ⊆ Rⁿ → Rⁿ, in the direction “r” is defined as:

∫_C F(r). dr =

∫_a^b

F[r(t)] . r’(t)dt.

Here, “.” represents the dot product.

Integrals of Particular Functions

Integral is defined as the method to sum the functions on a large scale. These integrals can be widely applied in real life, such as finding the area between curves, volume, the average value of the function, kinetic energy, center of mass, work-done, etc.

∫1/(x² – a²) dx = (1/2a) log|(x – a)/(x + a)| + C
∫1/(a² – x²) dx = (1/2a) log|(a + x)/(a – x)| + C
∫1/(x² + a²) dx = (1/a) tan^-1(x/a) + C
∫1/√(x² – a²) dx = log|x + √(x² – a²)| + C
∫1/√(a² – x²) dx = sin^-1(x/a) + C
∫1/√(x² + a²) dx = log|x + √(x² + a²)| + C

NCERT Solutions For Class 12 Maths Chapter 7 Exercises

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NCERT Solutions For Class 12 Mathematics Chapter 7 Integrals

Class 12 Maths NCERT Solutions Chapter 7 Integrals

NCERT Solutions Class 12 Mathematics Chapter 7 Important Topics

Double Integral

Continuous Integration

Properties of Definite Integral

Line Integral

Integrals of Particular Functions

NCERT Solutions For Class 12 Maths Chapter 7 Exercises

CBSE CLASS XII Related Questions

1.
For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?

2.
If A=\(\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\)verify that A³-6A²+9A-4 I=0 and hence find A^-1

3.
Find the following integral: \(\int (ax^2+bx+c)dx\)

4.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

5.
Solve system of linear equations, using matrix method.
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12

6.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)

Similar Mathematics Concepts

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NCERT Solutions For Class 12 Mathematics Chapter 7 Integrals

Class 12 Maths NCERT Solutions Chapter 7 Integrals

NCERT Solutions Class 12 Mathematics Chapter 7 Important Topics

Double Integral

Continuous Integration

Properties of Definite Integral

Line Integral

Integrals of Particular Functions

NCERT Solutions For Class 12 Maths Chapter 7 Exercises

CBSE CLASS XII Related Questions

1. For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?

2. If A=\(\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\)verify that A3-6A2+9A-4 I=0 and hence find A-1

3. Find the following integral: \(\int (ax^2+bx+c)dx\)

4. Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

5. Solve system of linear equations, using matrix method. x-y+2z=7 3x+4y-5z=-5 2x-y+3z=12

6. If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that (i) \((A+B)'=A'+B' \)(ii) \((A-B)'=A'-B'\)

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?

2.
If A=\(\begin{bmatrix}2&-1&1\\-1&2&-1\\1&-1&2\end{bmatrix}\)verify that A³-6A²+9A-4 I=0 and hence find A^-1

3.
Find the following integral: \(\int (ax^2+bx+c)dx\)

4.
Let f: R→R be defined as f(x) = 3x. Choose the correct answer.

5.
Solve system of linear equations, using matrix method.
x-y+2z=7
3x+4y-5z=-5
2x-y+3z=12

6.
If A'= \(\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 &1 \end{bmatrix}\)\(\begin{bmatrix} -1 & 2 & 1 \\ 1 &2 & 3\end{bmatrix}\) , then verify that
(i) \((A+B)'=A'+B' \)
(ii) \((A-B)'=A'-B'\)