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

## Solutions for Chapter 4: Definite Integration

Below listed, you can find solutions for Chapter 4 of Maharashtra State Board Balbharati for Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board.

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 : ""int_1^3 (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^(π/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^π 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 integrals : 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)]

## Solutions for 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 has the Maharashtra State Board Mathematics Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Maharashtra State Board solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. Balbharati solutions for Mathematics Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board Maharashtra State Board 4 (Definite Integration) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and provide excellent 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 Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board solutions Definite Integration exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in Balbharati Solutions are essential questions that can be asked in the final exam. Maximum Maharashtra State Board Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board students prefer Balbharati Textbook Solutions to score more in exams.

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

Share