Advertisement Remove all ads

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Definite Integration [Latest edition]

Chapters

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Definite Integration - Shaalaa.com
Advertisement Remove all ads
Advertisement Remove all ads

Chapter 4: Definite Integration

Exercise 4.1Exercise 4.2Miscellaneous Exercise 4
Exercise 4.1 [Page 156]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Definite Integration Exercise 4.1 [Page 156]

Exercise 4.1 | Q 1 | Page 156

Evaluate the following integrals as limit of a sum : `""^3int_1 (3x - 4)*dx`

Exercise 4.1 | Q 2 | Page 156

Evaluate the following integrals as limit of a sum : \[\int\limits_0^4 x^2 \cdot dx\]

Exercise 4.1 | Q 3 | Page 156

Evaluate the following integrals as limit of a sum : \[\int\limits_0^2 e^x\cdot dx\]

Exercise 4.1 | Q 4 | Page 156

Evaluate the following integrals as limit of a sum : \[\int\limits_0^2 (3x^2 - 1)\cdot dx\]

Exercise 4.1 | Q 5 | Page 156

Evaluate the following integrals as limit of a sum : \[\int\limits_1^3 x^3 \cdot dx\]

Exercise 4.2 [Pages 171 - 172]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Definite Integration Exercise 4.2 [Pages 171 - 172]

Exercise 4.2 | Q 1.01 | Page 171

Evaluate : `int_1^9(x + 1)/sqrt(x)*dx`

Exercise 4.2 | Q 1.02 | Page 171

Evaluate : `int_2^3 (1)/(x^2 + 5x + 6)*dx`

Exercise 4.2 | Q 1.03 | Page 171

Evaluate : `int_0^(pi/4) cot^2x*dx`

Exercise 4.2 | Q 1.04 | Page 171

Evaluate : `int_((-pi)/4)^(pi/4) (1)/(1 - sinx)*dx`

Exercise 4.2 | Q 1.05 | Page 171

Evaluate : `int_3^5 (1)/(sqrt(2x + 3) - sqrt(2x - 3))*dx`

Exercise 4.2 | Q 1.06 | Page 171

Evaluate : `int_0^1 (x^2 - 2)/(x^2 + 1)*dx`

Exercise 4.2 | Q 1.07 | Page 171

Evaluate : `int_0^(pi/4) sin 4x sin 3x *dx`

Exercise 4.2 | Q 1.08 | Page 171

Evaluate : `int_0^(pi/4) sqrt(1 + sin 2x)*dx`

Exercise 4.2 | Q 1.09 | Page 171

Evaluate : `int_0^(pi/4) sin^4x*dx`

Exercise 4.2 | Q 1.1 | Page 171

Evaluate : `int_(-4)^2 (1)/(x^2 + 4x + 13)*dx`

Exercise 4.2 | Q 1.11 | Page 171

Evaluate : `int_0^4 (1)/sqrt(4x - x^2)*dx`

Exercise 4.2 | Q 1.12 | Page 171

Evaluate : `int_0^1 (1)/sqrt(3 + 2x - x^2)*dx`

Exercise 4.2 | Q 1.13 | Page 171

Evaluate : `int_0^(pi/2) x sin x*dx`

Exercise 4.2 | Q 1.14 | Page 171

Evaluate : `int_0^1 x tan^-1x*dx`

Exercise 4.2 | Q 1.15 | Page 171

Evaluate : `int_0^oo xe^-x *dx`

Exercise 4.2 | Q 2.01 | Page 172

Evaluate : `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2)*dx`

Exercise 4.2 | Q 2.02 | Page 172

Evaluate : `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x +1)*dx`

Exercise 4.2 | Q 2.03 | Page 172

Evaluate : `int_0^(pi/4) (sin2x)/(sin^4x + cos^4x)*dx`

Exercise 4.2 | Q 2.04 | Page 172

Evaluate : `int_0^(2pi) sqrt(cos x) sin^3x*dx`

Exercise 4.2 | Q 2.05 | Page 172

Evaluate : `int_0^(pi/2) (1)/(5 + 4 cos x)*dx`

Exercise 4.2 | Q 2.06 | Page 172

Evaluate : `int_0^(pi/4) (cosx)/(4 - sin^2x)*dx`

Exercise 4.2 | Q 2.07 | Page 172

Evaluate : `int_0^(pi/2) cosx/((1 + sinx)(2 + sin x))*dx`

Exercise 4.2 | Q 2.08 | Page 172

Evaluate : `int_(-1)^1 (1)/(a^2e^x + b^2e^(-x))*dx`

Exercise 4.2 | Q 2.09 | Page 172

Evaluate : `int_0^pi (1)/(3 + 2sinx + cosx)*dx`

Exercise 4.2 | Q 2.1 | Page 172

Evaluate : `int_0^(pi/4) sec^4x*dx`

Exercise 4.2 | Q 2.11 | Page 172

Evaluate : `int_0^1 sqrt((1 - x)/(1 + x))*dx`

Exercise 4.2 | Q 2.12 | Page 172

Evaluate : `int_0^pi sin^3x (1 + 2cosx)(1 + cosx)^2*dx`

Exercise 4.2 | Q 2.13 | Page 172

Evaluate : `int_0^(pi/2) sin2x*tan^-1 (sinx)*dx`

Exercise 4.2 | Q 2.14 | Page 172

Evaluate : `int _((1)/(sqrt(2)))^1 (e^(cos^-1x) sin^-1x)/(sqrt(1 - x^2))*dx`

Exercise 4.2 | Q 2.15 | Page 172

Evaluate : `int_1^3 (cos(logx))/x*dx`

Exercise 4.2 | Q 3.01 | Page 172

Evaluate the following: `int_0^a (1)/(x + sqrt(a^2 - x^2))*dx`

Exercise 4.2 | Q 3.02 | Page 172

Evaluate the following : `int_0^(pi/2) log(tanx)*dx`

Exercise 4.2 | Q 3.03 | Page 172

Evaluate the following : `int_0^1 log(1/x - 1)*dx`

Exercise 4.2 | Q 3.04 | Page 172

Evaluate : `int_0^(pi/2) (sinx - cosx)/(1 + sinx cosx)*dx`

Exercise 4.2 | Q 3.05 | Page 172

Evaluate the following : `int_0^3 x^2(3 - x)^(5/2)*dx`

Exercise 4.2 | Q 3.06 | Page 172

Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`

Exercise 4.2 | Q 3.07 | Page 172

Evaluate the following : `int_((-pi)/2)^(pi/2) log((2 + sinx)/(2 - sinx))*dx`

Exercise 4.2 | Q 3.08 | Page 172

Evaluate the following :  `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`

Exercise 4.2 | Q 3.09 | Page 172

Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`

Exercise 4.2 | Q 3.1 | Page 172

Evaluate the following : `int_0^1 (log(x + 1))/(x^2 + 1)*dx`

Exercise 4.2 | Q 3.11 | Page 172

Evaluate the following : `int_(-1)^(1) (x^3 + 2)/sqrt(x^2 + 4)*dx`

Exercise 4.2 | Q 3.12 | Page 172

Evaluate the following : `int_(-a)^(a) (x + x^3)/(16 - x^2)*dx`

Exercise 4.2 | Q 3.13 | Page 172

Evaluate the following : `int_0^1 t^2 sqrt(1 - t)*dt`

Exercise 4.2 | Q 3.14 | Page 172

Evaluate the following : `int_0^pi x sin x cos^2x*dx`

Exercise 4.2 | Q 3.15 | Page 172

Evaluate the following : `int_0^1 (logx)/sqrt(1 - x^2)*dx`

Miscellaneous Exercise 4 [Pages 175 - 177]

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Chapter 4 Definite Integration Miscellaneous Exercise 4 [Pages 175 - 177]

Miscellaneous Exercise 4 | Q 1.01 | Page 175

Choose the correct option from the given alternatives :

`int_2^3 dx/(x(x^3 - 1))` =

  • `(1)/(3) log (208/189)`

  • `(1)/(3) log (189/208)`

  • `log (208/189)`

  • `log (189/208)`

Miscellaneous Exercise 4 | Q 1.02 | Page 175

Choose the correct option from the given alternatives : 

`int_0^(pi/2) (sin^2x*dx)/(1 + cosx)^2` =

  • `(4 - pi)/2`

  • `(pi - 4)/2`

  • `4 - pi/(2)`

  • `(4 + pi)/2`

Miscellaneous Exercise 4 | Q 1.03 | Page 175

Choose the correct option from the given alternatives :

`int_0^(log5) (e^x sqrt(e^x - 1))/(e^x + 3)*dx` =

  • 3 + 2π

  • 2 + π

  • 4 – π

  • 4 + π

Miscellaneous Exercise 4 | Q 1.04 | Page 175

Choose the correct option from the given alternatives : 

`int_0^(pi/2) sn^6x cos^2x*dx` =

  • `(7pi)/(256)`

  • `(3pi)/(256)`

  • `(5pi)/(256)`

  • `(-5pi)/(256)`

Miscellaneous Exercise 4 | Q 1.05 | Page 175

Choose the correct option from the given alternatives : 

If `dx/(sqrt(1 + x) - sqrt(x)) = k/(3)`, then k is equal to

  • `sqrt(2)(2sqrt(2) - 2)`

  • `sqrt(2)/(3)(2 - 2sqrt(2))`

  • `(2sqrt(2) - 2)/(3)`

  • `4sqrt(2)`

Miscellaneous Exercise 4 | Q 1.06 | Page 175

Choose the correct option from the given alternatives : 

`int_1^2 (1)/x^2 e^(1/x)*dx` = 

  • `sqrt(e) + 1`

  • `sqrt(e) - 1`

  • `sqrt(e)(sqrt(e) - 1)`

  • `(sqrt(e) - 1)/e`

Miscellaneous Exercise 4 | Q 1.07 | Page 175

Choose the correct option from the given alternatives :

If `[1/logx - 1/(logx)^2]*dx = a + b/(log2)`, then

  • a = e, b = – 2

  • a = e, b = 2

  • a = – e, b = 2

  • a = – e, b = – 2

Miscellaneous Exercise 4 | Q 1.08 | Page 175

Choose the correct option from the given alternatives :

Let I1 = `int_e^(e^2) dx/logx  "and"  "I"_2 = int_1^2 e^x/x*dx`, then

  • I1 = `(1)/(3)"I"_2`

  • I1 + I = 0

  • I1 = 2I 

  • I1 = I 

Miscellaneous Exercise 4 | Q 1.09 | Page 176

Choose the correct option from the given alternatives :

`int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =

  • 9

  • `(9)/(2)`

  • 0

  • 1

Miscellaneous Exercise 4 | Q 1.1 | Page 176

Choose the correct option from the given alternatives :

The value of `int_((-pi)/4)^(pi/4) log((2+ sin theta)/(2 - sin theta))*d theta` is

  • 0

  • 1

  • 2

  • `pi`

Miscellaneous Exercise 4 | Q 2.01 | Page 176

Evaluate the following : `int_0^(pi/2) cosx/(3cosx + sinx)*dx`

Miscellaneous Exercise 4 | Q 2.02 | Page 176

Evaluate the following : `int_(pi/4)^(pi/2) (cos theta)/[cos  theta/2 + sin  theta/2]^3*d theta`

Miscellaneous Exercise 4 | Q 2.03 | Page 176

Evaluate the following : `int_0^1 1/(1 + sqrt(x))*dx`

Miscellaneous Exercise 4 | Q 2.04 | Page 176

Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`

Miscellaneous Exercise 4 | Q 2.05 | Page 176

Evaluate the following : `int_0^1 t^5 sqrt(1 - t^2)*dt`

Miscellaneous Exercise 4 | Q 2.06 | Page 176

Evaluate the following : `int_0^1 (cos^-1 x^2)*dx`

Miscellaneous Exercise 4 | Q 2.07 | Page 176

Evaluate the following : `int_(-1)^(1) (1 + x^3)/(9 - x^2)*dx`

Miscellaneous Exercise 4 | Q 2.08 | Page 176

Evaluate the following : `int_0^pi x*sinx*cos^4x*dx`

Miscellaneous Exercise 4 | Q 2.09 | Page 176

Evaluate the following : `int_0^pi x/(1 + sin^2x)*dx`

Miscellaneous Exercise 4 | Q 2.1 | Page 176

Evaluate the following : `int_1^oo 1/(sqrt(x)(1 + x))*dx`

Miscellaneous Exercise 4 | Q 3.01 | Page 176

Evaluate the following : `int_0^1 (1/(1 + x^2))sin^-1((2x)/(1 + x^2))*dx`

Miscellaneous Exercise 4 | Q 3.02 | Page 176

Evaluate the following : `int_0^(pi/2) 1/(6 - cosx)*dx`

Miscellaneous Exercise 4 | Q 3.03 | Page 176

Evaluate the following : `int_0^a 1/(a^2 + ax - x^2)*dx`

Miscellaneous Exercise 4 | Q 3.04 | Page 176

Evaluate the following : `int_(pi/5)^((3pi)/10) sinx/(sinx + cosx)*dx`

Miscellaneous Exercise 4 | Q 3.05 | Page 176

Evaluate the following : `int_0^1 sin^-1 ((2x)/(1 + x^2))*dx`

Miscellaneous Exercise 4 | Q 3.06 | Page 176

Evaluate the following : `int_0^(pi/4) (cos2x)/(1 + cos 2x + sin 2x)*dx`

Miscellaneous Exercise 4 | Q 3.07 | Page 176

Evaluate the following : `int_0^(pi/2) [2 log (sinx) - log (sin 2x)]*dx`

Miscellaneous Exercise 4 | Q 3.08 | Page 176

Evaluate the following : `int_0^pi  (sin^-1x + cos^-1x)^3 sin^3x*dx`

Miscellaneous Exercise 4 | Q 3.09 | Page 176

Evaluate the following : `int_0^4 [sqrt(x^2 + 2x + 3]]^-1*dx`

Miscellaneous Exercise 4 | Q 3.1 | Page 176

Evaluate the following : `int_(-2)^(3) |x - 2|*dx`

Miscellaneous Exercise 4 | Q 4.1 | Page 177

Evaluate the following : if `int_a^a sqrt(x)*dx = 2a int_0^(pi/2) sin^3x*dx`, find the value of `int_a^(a + 1)x*dx`

Miscellaneous Exercise 4 | Q 4.2 | Page 177

Evaluate the following : If `int_0^k 1/(2 + 8x^2)*dx = pi/(16)`, find k

Miscellaneous Exercise 4 | Q 4.3 | Page 177

Evaluate the following : If f(x) = a + bx + cx2, show that `int_0^1 f(x)*dx = (1/(6)[f(0) + 4f(1/2) + f(1)]`

Advertisement Remove all ads

Chapter 4: Definite Integration

Exercise 4.1Exercise 4.2Miscellaneous Exercise 4
Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Definite Integration - Shaalaa.com

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 - Definite Integration

Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 (Definite Integration) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board chapter 4 Definite Integration are Definite Integral as Limit of Sum, Fundamental Theorem of Integral Calculus, Methods of Evaluation and Properties of Definite Integral.

Using Balbharati 12th Board Exam solutions Definite Integration exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board 12th Board Exam prefer Balbharati Textbook Solutions to score more in exam.

Get the free view of chapter 4 Definite Integration 12th Board Exam extra questions for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board and can use Shaalaa.com to keep it handy for your exam preparation

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×