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

## 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)]

## 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

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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.

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