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Chapters
1.2: Matrics
1.3: Trigonometric Functions
1.4: Pair of Lines
1.5: Vectors and Three Dimensional Geometry
1.6: Line and Plane
1.7: Linear Programming Problems
2.1: Differentiation
2.2: Applications of Derivatives
2.3: Indefinite Integration
▶ 2.4: Definite Integration
2.5: Application of Definite Integration
2.6: Differential Equations
2.7: Probability Distributions
2.8: Binomial Distribution
![SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.4 - Definite Integration - Shaalaa.com](/images/mathematics-and-statistics-arts-and-science-english-12-standard-hsc_6:5f2b1b2038084cf381bfa42c826a928c.jpg "SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.4 - Definite Integration")
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Solutions for Chapter 2.4: Definite Integration
Below listed, you can find solutions for Chapter 2.4 of Maharashtra State Board SCERT Maharashtra for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC.
SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC 2.4 Definite Integration MCQ
2 Marks each
`int_1^9 (x + 1)/sqrt(x) "d"x` =
`80/3`
`64/3`
`17/3`
`15/3`
`int_0^1 sqrt((1 - x)/(1 + x)) "d"x` =
`pi/2 - 1`
`pi/2 + 1`
`pi/2 - 2`
`p/2 + 2`
`int_1^2 ("e"^(1/x))/(x^2) "d"x` =
`2sqrt("e")(1 + sqrt("e"))`
`sqrt("e")(1 - sqrt("e"))`
`sqrt("e")(sqrt("e") - 1)`
`sqrt("e")(1 + sqrt("e"))`
`int_0^(x/4) sqrt(1 + sin 2x) "d"x` =
`1/sqrt(2)`
`sqrt(2) + 1`
`2sqrt(2)`
1
If `int_0^1 ("d"x)/(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)`
`int_(pi/5)^((3pi)/10) sinx/(sinx + cosx) "d"x` =
`pi/10`
`pi/20`
`pi/6`
`pi/12`
`int_0^1 (x^2 - 2)/(x^2 + 1) "d"x` =
`1 - (3pi)/4`
`2 - (3pi)/4`
`1 + (3pi)/4`
`2 + (3pi)/4`
Let I1 = `int_"e"^("e"^2) 1/logx "d"x` and I2 = `int_1^2 ("e"^x)/x "d"x` then
I1 = `1/3 "I"_2`
I1 + I2 = 0
I1 = 2I2
I1 = I2
`int_0^4 1/sqrt(4x - x^2) "d"x` =
0
2π
π
4π
`int_0^(pi/2) log(tanx) "d"x` =
`pi/8(log2)`
0
`- pi/8 (log2)`
`pi/2 (log2)`
SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC 2.4 Definite Integration Very Short Answers
1 Mark
Evaluate: `int_(pi/6)^(pi/3) cosx "d"x`
Evaluate: `int_(- pi/4)^(pi/4) x^3 sin^4x "d"x`
Evaluate: `int_0^1 1/(1 + x^2) "d"x`
Evaluate: `int_0^(pi/4) sec^2 x "d"x`
Evaluate: `int_0^1 |x| "d"x`
Evaluate: `int_0^1 1/sqrt(1 - x^2) "d"x`
Evaluate: `int_1^2 x/(1 + x^2) "d"x`
Evaluate: `int_0^1 "e"^x/sqrt("e"^x - 1) "d"x`
Evaluate: `int_0^(pi/2) (sin2x)/(1 + sin^2x) "d"x`
Evaluate: `int_0^1(x + 1)^2 "d"x`
SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC 2.4 Definite Integration Short Answers I
2 Marks
Evaluate: `int_(pi/6)^(pi/3) sin^2 x "d"x`
Evaluate: `int_0^(pi/2) sqrt(1 - cos 4x) "d"x`
Evaluate: `int_0^(pi/2) cos^3x "d"x`
Evaluate: `int_0^pi cos^2 x "d"x`
Evaluate: `int_0^(pi/4) (tan^3x)/(1 + cos 2x) "d"x`
Evaluate: `int_0^(pi/4) cosx/(4 - sin^2 x) "d"x`
Evaluate: `int_1^3 (cos(logx))/x "d"x`
Evaluate: `int_0^(pi/2) (sin^2x)/(1 + cos x)^2 "d"x`
Evaluate: `int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x) "d"x`
SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC 2.4 Definite Integration Short Answers II
3 Marks
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"c""f"(x) "d"x + int_"c"^"b" "f"(x) "d"x`, where a < c < b
Prove that: `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`
Prove that: `int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x`. Hence find `int_0^(pi/2) sin^2x "d"x`
Evaluate: `int_0^(pi/2) (sin^4x)/(sin^4x + cos^4x) "d"x`
Evaluate: `int_3^8 (11 - x)^2/(x^2 + (11 - x)^2) "d"x`
Evaluate: `int_(-1)^1 |5x - 3| "d"x`
Evaluate: `int_(-4)^2 1/(x^2 + 4x + 13) "d"x`
Evaluate: `int_0^1 1/sqrt(3 + 2x - x^2) "d"x`
Evaluate: `int_0^1 x* tan^-1x "d"x`
Evaluate: `int_0^(1/sqrt(2)) (sin^-1x)/(1 - x^2)^(3/2) "d"x`
Evaluate: `int_0^(pi/4) sec^4x "d"x`
Evaluate: `int_0^(pi/2) 1/(5 + 4cos x) "d"x`
Evaluate: `int_0^(pi/2) cos x/((1 + sinx)(2 + sinx)) "d"x`
Evaluate: `int_(-1)^1 1/("a"^2"e"^x + "b"^2"e"^(-x)) "d"x`
Evaluate: `int_0^"a" 1/(x + sqrt("a"^2 - x^2)) "d"x`
Evaluate: `int_0^3 x^2 (3 - x)^(5/2) "d"x`
Evaluate: `int_0^1 "t"^2 sqrt(1 - "t") "dt"`
SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC 2.4 Definite Integration Long Answers III
4 Marks
Prove that: `int_0^(2"a") "f"(x) "d"x = int_0^"a" "f"(x) "d"x + int_0^"a" "f"(2"a" - x) "d"x`
Prove that:
`{:(int_(-a)^a f(x) dx = 2 int_0^a f(x) dx",", "If" f(x) "is an even function"),( = 0",", "if" f(x) "is an odd function"):}`
Evaluate: `int_0^(1/2) 1/((1 - 2x^2) sqrt(1 - x^2)) "d"x`
Evaluate: `int_0^(pi/4) (sec^2x)/(3tan^2x + 4tan x + 1) "d"x`
Evaluate: `int_(1/sqrt(2))^1 (("e"^(cos^-1x))(sin^-1x))/sqrt(1 - x^2) "d"x`
Evaluate: `int_0^1 (log(x + 1))/(x^2 + 1) "d"x`
Evaluate: `int_0^pi x*sinx*cos^2x* "d"x`
Evaluate: `int_0^(pi/2) x sin x.dx`
Evaluate: `int_(-1)^1 (1 + x^2)/(9 - x^2) "d"x`
Evaluate: `int_0^1 (1/(1 + x^2)) sin^-1 ((2x)/(1 + x^2)) "d"x`
Evaluate: `int_0^(pi/4) (cos2x)/(1 + cos 2x + sin 2x) "d"x`
Evaluate: `int_0^(pi/4) log(1 + tanx) "d"x`
Evaluate: `int_0^pi 1/(3 + 2sinx + cosx) "d"x`
Solutions for 2.4: Definite Integration
SCERT Maharashtra solutions for Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.4 - Definite Integration
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Concepts covered in Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC chapter 2.4 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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