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Chapters
▶ 2: Integral Calculus – 1
3: Integral Calculus – 2
4: Differential Equations
5: Numerical Methods
6: Random Variable and Mathematical expectation
7: Probability Distributions
8: Sampling techniques and Statistical Inference
9: Applied Statistics
10: Operations Research
![Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board chapter 2 - Integral Calculus – 1 - Shaalaa.com](/images/business-mathematics-and-statistics-english-class-12-tn-board_6:5f2b1b2038084cf381bfa42c826a928c.jpg "Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board chapter 2 - Integral Calculus – 1")
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Solutions for Chapter 2: Integral Calculus – 1
Below listed, you can find solutions for Chapter 2 of Tamil Nadu Board of Secondary Education Samacheer Kalvi for Business Mathematics and Statistics [English] Class 12 TN Board.
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.1 [Page 30]
Integrate the following with respect to x.
`sqrt(3x + 5)`
Integrate the following with respect to x.
`(9x^2 - 4/x^2)^2`
Integrate the following with respect to x.
(3 + x)(2 – 5x)
Integrate the following with respect to x.
`sqrt(x)(x^3 - 2x + 3)`
Integrate the following with respect to x.
`(8x + 13)/sqrt(4x + 7)`
Integrate the following with respect to x.
`1/(sqrt(x + 1) + sqrt(x - 1))`
Integrate the following with respect to x.
If f'(x) = x + b, f(1) = 5 and f(2) = 13, then find f(x)
Integrate the following with respect to x.
If f'(x) = 8x3 – 2x and f(2) = 8, then find f(x)
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.2 [Page 31]
Integrate the following with respect to x.
`(sqrt(2x) - 1/sqrt(2x))^2`
Integrate the following with respect to x.
`(x^4 - x^2 + 2)/(x - 1)`
Integrate the following with respect to x.
`x^3/(x + 2)`
Integrate the following with respect to x.
`(x^3 + 3x^2 - 7x + 11)/(x + 5)`
Integrate the following with respect to x.
`(3x + 2)/((x - 2)(x - 3))`
Integrate the following with respect to x.
`(4x^2 + 2x + 6)/((x + 1)^2(x - 3))`
Integrate the following with respect to x.
`(3x^2 - 2x + 5)/((x - 1)(x^2 + 5))`
Integrate the following with respect to x.
If f'(x) = `1/x` and f(1) = `pi/4`, then find f(x)
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.3 [Page 32]
Integrate the following with respect to x.
`"e"^(xlog"a") + "e"^("a"log"a") - "e"^("n"logx)`
Integrate the following with respect to x.
`("a"^x - "e"^(xlog"b"))/("e"^(x log "a") "b"^x)`
Integrate the following with respect to x.
`("e"^x + 1)^2 "e"^x`
Integrate the following with respect to x.
`("e"^(3x) - "e"^(-3x))/"e"^x`
Integrate the following with respect to x.
`("e"^(3x) +"e"^(5x))/("e"^x + "e"^-x)`
Integrate the following with respect to x.
`[1 - 1/2]"e"^((x + 1/x))`
Integrate the following with respect to x.
`1/(x(log x)^2`
Integrate the following with respect to x.
If f'(x) = ex and f(0) = 2, then find f(x)
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.4 [Page 33]
Integrate the following with respect to x.
2 cos x – 3 sin x + 4 sec2x – 5 cosec2x
Integrate the following with respect to x.
sin3x
Integrate the following with respect to x.
`(cos 2x + 2sin^2x)/(cos^2x)`
Integrate the following with respect to x.
`1/(sin^2x cos^2x) ["Hint:" sin^2x + cos^2x = 1]`
Integrate the following with respect to x.
`sqrt(1 - sin 2x)`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.5 [Page 35]
Integrate the following with respect to x.
xe–x
Integrate the following with respect to x.
x3e3x
Integrate the following with respect to x.
log x
Integrate the following with respect to x.
x log x
Integrate the following with respect to x.
xn log x
Integrate the following with respect to x.
`x^5 "e"^x`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.6 [Page 38]
Integrate the following with respect to x.
`(2x + 5)/(x^2 + 5x - 7)`
Integrate the following with respect to x.
`("e"^(3logx))/(x^4 + 1)`
Integrate the following with respect to x.
`"e"^(2x)/("e"^(2x) - 2)`
Integrate the following with respect to x.
`(log x)^3/x`
Integrate the following with respect to x.
`(6x + 7)/sqrt(3x^2 + 7x - 1)`
Integrate the following with respect to x.
`(4x + 2) sqrt(x^2 + x + 1)`
Integrate the following with respect to x.
x8(1 + x9)5
Integrate the following with respect to x.
`(x^("e" - 1) + "e"^(x - 1))/(x^"e" + "e"^x)`
Integrate the following with respect to x
`1/(x log x)`
Integrate the following with respect to x.
`x/(2x^4 - 3x^2 - 2)`
Integrate the following with respect to x.
ex(1 + x) log(xex)
Integrate the following with respect to x.
`1/(x^2(x^2 + 1))`
Integrate the following with respect to x.
`"e"^x [1/x^2 - 2/x^3]`
Integrate the following with respect to x.
`"e"^x [(x - 1)/(x + 1)^3]`
Integrate the following with respect to x.
`"e"^(3x) [(3x - 1)/(9x^2)]`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.7 [Page 42]
Integrate the following with respect to x.
`1/(9 - 16x^2)`
Integrate the following with respect to x.
`1/(9 - 8x - x^2)`
Integrate the following with respect to x.
`1/(2x^2 - 9)`
Integrate the following with respect to x.
`1/(x^2 - x - 2)`
Integrate the following with respect to x.
`1/(x^2 + 3x + 2)`
Integrate the following with respect to x.
`1/(2x^2 + 6x - 8)`
Integrate the following with respect to x.
`"e"^x/("e"^(2x) - 9)`
Integrate the following with respect to x.
`1/sqrt(9x^2 - 7)`
Integrate the following with respect to x.
`1/sqrt(x^2 + 6x + 13)`
Integrate the following with respect to x.
`1/sqrt(x^2 - 3x + 2)`
Integrate the following with respect to x.
`x^3/sqrt(x^8 - 1)`
Integrate the following with respect to x.
`sqrt(1 + x + x^2)`
Integrate the following with respect to x.
`sqrt(x^2 - 2)`
Integrate the following with respect to x.
`sqrt(4x^2 - 5)`
Integrate the following with respect to x.
`sqrt(2x^2 + 4x + 1)`
Integrate the following with respect to x.
`1/(x + sqrt(x^2 - 1)`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.8 [Page 47]
Using second fundamental theorem, evaluate the following:
`int_0^1 "e"^(2x) "d"x`
Using second fundamental theorem, evaluate the following:
`int_0^(1/4) sqrt(1 - 4) "d"x`
Using second fundamental theorem, evaluate the following:
`int_1^2 (x "d"x)/(x^2 + 1)`
Using second fundamental theorem, evaluate the following:
`int_0^3 ("e"^x "d"x)/(1 + "e"^x)`
Using second fundamental theorem, evaluate the following:
`int_0^1 x"e"^(x^2) "d"x`
Using second fundamental theorem, evaluate the following:
`int_1^"e" ("d"x)/(x(1 + logx)^3`
Using second fundamental theorem, evaluate the following:
`int_(-1)^1 (2x + 3)/(x^2 + 3x + 7) "d"x`
Using second fundamental theorem, evaluate the following:
`int_0^(pi/2) sqrt(1 + cos x) "d"x`
Using second fundamental theorem, evaluate the following:
`int_1^2 (x - 1)/x^2 "d"x`
Evaluate the following:
`int_1^4` f(x) dx where f(x) = `{{:(4x + 3",", 1 ≤ x ≤ 2),(3x + 5",", 2 < x ≤ 4):}`
Evaluate the following:
`int_0^2 "f"(x) "d"x` where f(x) = `{{:(3 - 2x - x^2",", x ≤ 1),(x^2 + 2x - 3",", 1 < x ≤ 2):}`
Evaluate the following:
`int_(-1)^1 "f"(x) "d"x` where f(x) = `{{:(x",", x ≥ 0),(-x",", x < 0):}`
Evaluate the following:
f(x) = `{{:("c"x",", 0 < x < 1),(0",", "otherwise"):}` Find 'c" if `int_0^1 "f"(x) "d"x` = 2
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.9 [Page 50]
Evaluate the following using properties of definite integral:
`int_(- pi/4)^(pi/4) x^3 cos^3 x "d"x`
Evaluate the following using properties of definite integral:
`int_(- pi/2)^(pi/2) sin^2theta "d"theta`
Evaluate the following using properties of definite integral:
`int_(-1)^1 log ((2 - x)/(2 + x)) "d"x`
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
Evaluate the following using properties of definite integral:
`int_0^1 log (1/x - 1) "d"x`
Evaluate the following using properties of definite integral:
`int_0^1 x/((1 - x)^(3/4)) "d"x`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.10 [Page 51]
Evaluate the following:
Γ(4)
Evaluate the following:
`Γ (9/2)`
Evaluate the following:
`int_0^oo "e"^(-mx) x^6 "d"x`
Evaluate the following:
`int_0^oo "e"^(-4x) x^4 "d"x`
Evaluate the following:
`int_0^oo "e"^(- x/2) x^5 "d"x`
If f(x) = `{{:(x^2"e"^(-2x)",", x ≥ 0),(0",", "otherwise"):}`, then evaluate `int_0^oo "f"(x) "d"x`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.11 [Page 53]
Evaluate the following integrals as the limit of the sum:
`int_0^1 (x + 4) "d"x`
Evaluate the following integrals as the limit of the sum:
`int_1^3 x "d"x`
Evaluate the following integrals as the limit of the sum:
`int_1^3 (2x + 3) "d"x`
Evaluate the following integrals as the limit of the sum:
`int_0^1 x^2 "d"x`
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Exercise 2.12 [Pages 53 - 55]
MCQ
Choose the correct alternative:
`int 1/x^3 "d"x` is
`(-3)/(x^2) + "c"`
`(-1)/(2x^2) + "c"`
`(-1)/(3x^2) + "c"`
`(-2)/(x^2) + "c"`
Choose the correct alternative:
`int 2^x "d"x` is
2x log 2 + c
2x + c
`2^x/log2 + "c"`
`log2/2^x + "c"`
Choose the correct alternative:
`int (sin2x)/(2sinx) "d"x` is
sin x + c
`1/2 sin x + "c"`
`cos x + "c"`
`1/2 cos x + "c"`
Choose the correct alternative:
`int (sin5x - sinx)/(cos3x) "d"x` is
− cos 2x + c
− cos 2x + c
`- 1/4 cos 2x + "c"`
− 4 cos 2x + c
Choose the correct alternative:
`int logx/x "d"x, x > 0` is
`1/2 (log x)^2 + "c"`
`- 1/2 (log x)^2`
`2/x^2 + "c"`
`2/x^2 - "c"`
Choose the correct alternative:
`int "e"^x/sqrt(1 + "e"^x) "d"x` is
`"e"^x/sqrt(1 + "e"^x) + "c"`
`2sqrt(1 + "e"^x) + "c"`
`sqrt(1 + "e"^x) + "c"`
`"e"^x sqrt(1 + "e"^x) + "c"`
Choose the correct alternative:
`int sqrt("e"^x) "d"x` is
`sqrt("e"^x) + "c"`
`2sqrt("e"^x) + "c"`
`1/2 sqrt("e"^x) + "c"`
`1/(2sqrt("e"^x)) + "c"`
Choose the correct alternative:
`int "e"^(2x) [2x^2 + 2x] "d"x`
`"e"^(2x)x^2 + "c"`
`x"e"^(2x) + "c"`
`2x^2"e"^2 + "c"`
`(x^2"e"^x)/2 + "c"`
Choose the correct alternative:
`int "e"^x/("e"^x + 1) "d"x` is
`log|"e"^x/("e"^x + 1)| + "c"`
`log|("e"^x + 1)/"e"| + "c"`
`log|"e"^x| + "c"`
`log|"e"^x + 1| + "c"`
Choose the correct alternative:
`int[9/(x - 3) - 1/(x + 1)] "d"x` is
`log |x - 3| - log |x + 1| + "c"`
`log |x - 3| + log |x + 1| + "c"`
`9log |x - 3| - log |x + 1| + "c"`
`9log |x - 3| + log |x + 1| + "c"`
Choose the correct alternative:
`int (2x^3)/(4 + x^4) "d"x` is
`log |4 + x^4| + "c"`
`1/2 log |4 + x^4| + "c"`
`1/4 log |4 + x^4| + "c"`
`log |2x^3/(4 + x^4) + "c"`
Choose the correct alternative:
`int ("d"x)/sqrt(x^2 - 36) + "c"`
`sqrt(x^2 - 36) + "c"`
`log |x + sqrt(x^2 - 36)| + "c"`
`log |x - sqrt(x^2 - 36)| + "c"`
`log |x^2 + sqrt(x^2 - 36)| + "c"`
Choose the correct alternative:
`int (2x + 3)/sqrt(x^2 + 3x + 2) "d"x` is
`sqrt(x^2 + 3x + 2) + "c"`
`2sqrt(x^2 + 3x + 2) + "c"`
`log(x^2 + 3x + 2) + "c"`
`2/3(x^2 + 3x + 2) + "c"`
Choose the correct alternative:
`int_0^1 (2x + 1) "d"x` is
1
2
3
4
Choose the correct alternative:
`int_2^4 ("d"x)/x` is
log 4
0
log 2
log 8
Choose the correct alternative:
`int_0^oo "e"^(-2x) "d"x` is
0
1
2
`1/2`
Choose the correct alternative:
`int_(-1)^1 x^3 "e"^(x^4) "d"x` is
1
`2 int_0^1 x^3 "e"^(x^4) "d"x`
0
`"e"^(x^4)`
Choose the correct alternative:
If f(x) is a continuous function and a < c < b, then `int_"a"^"c" f(x) "d"x + int_"c"^"b" f(x) "d"x` is
`int_"a"^"b" f(x) "d"x - int_"a"^"c" f(x) "d"x`
`int_"a"^"c" f(x) "d"x - int_"a"^"b" f(x) "d"x`
`int_"a"^"b" f(x) "d"x`
0
Choose the correct alternative:
The value of `int_(- pi/2)^(pi/2) cos x "d"x` is
0
2
1
4
Choose the correct alternative:
`int_0^1 sqrt(x^4 (1 - x)^2) "d"x` is
`1/12`
`(-7)/12`
`7/12`
`(-1)/12`
Choose the correct alternative:
If `int_0^1 f(x) "d"x = 1, int_0^1 x f(x) "d"x = "a"`, and `int_0^1 x^2 f(x) "d"x = "a"^2`, then `int_0^1 ("a" - x)^2 f(x) "d"x` is
4a2
0
2a2
1
Choose the correct alternative:
The value of `int_2^3 f(5 - 3) "d"x - int_2^3 f(x) "d"x` is
1
0
– 1
5
Choose the correct alternative:
`int_0^4 (sqrt(x) + 1/sqrt(x)) "d"x` is
`20/3`
`21/3`
`28/3`
`1/3`
Choose the correct alternative:
`int_0^(pi/3) tan x "d"x` is
log 2
0
`log sqrt(2)`
2 log 2
Choose the correct alternative:
Using the factorial representation of the gamma function, which of the following is the solution for the gamma function Γ(n) when n = 8 is
5040
5400
4500
5540
Choose the correct alternative:
Γ(n) is
(n – 1)!
n!
n Γ(n)
(n – 1) Γ(n)
Choose the correct alternative:
Γ(1) is
0
1
n
n!
Choose the correct alternative:
If n > 0, then Γ(n) is
`int_0^1 "e"^-x x^("n" - 1) "d"x`
`int_0^1 "e"^-x x^"n" "d"x`
`int_0^oo "e"^x x^-"n" "d"x`
`int_0^oo "e"^-x x^("n" - 1) "d"x`
Choose the correct alternative:
`Γ(3/2)`
`sqrt(pi)`
`sqrt(pi)/2`
`2sqrt(pi)`
`3/2`
Choose the correct alternative:
`int_0^oo x^4"e"^-x "d"x` is
12
4
4!
64
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board 2 Integral Calculus – 1 Miscellaneous problems [Page 55]
Evaluate the following integral:
`int 1/(sqrt(x + 2) - sqrt(x + 3)) "d"x`
Evaluate the following integral:
`int ("d"x)/(2 - 3x - 2x^2)`
Evaluate the following integral:
`int ("d"x)/("e"^x + 6 + 5"e"^-x)`
Evaluate the following integral:
`int sqrt(2x^2 - 3) "d"x`
Evaluate the following integral:
`sqrt(9x^2 + 12x + 3) "d"x`
Evaluate the following integral:
`int (x + 1)^2 log x "d"x`
Evaluate the following integral:
`int log (x - sqrt(x^2 - 1)) "d"x`
Evaluate the following integral:
`int_0^1 sqrt(x(x - 1)) "d"x`
Evaluate the following integral:
`int_(-1)^1 x^2 "e"^(-2x) "d"x`
Evaluate the following integral:
`int_0^3 (x dx)/(sqrt(x + 1)+ sqrt(5x + 1))`
Solutions for 2: Integral Calculus – 1
Samacheer Kalvi solutions for Business Mathematics and Statistics [English] Class 12 TN Board chapter 2 - Integral Calculus – 1
Shaalaa.com has the Tamil Nadu Board of Secondary Education Mathematics Business Mathematics and Statistics [English] Class 12 TN Board Tamil Nadu Board of Secondary Education 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. Samacheer Kalvi solutions for Mathematics Business Mathematics and Statistics [English] Class 12 TN Board Tamil Nadu Board of Secondary Education 2 (Integral Calculus – 1) 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.
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Concepts covered in Business Mathematics and Statistics [English] Class 12 TN Board chapter 2 Integral Calculus – 1 are Indefinite Integration with Standard Indefinite Integral Formulae, Definite Integrals.
Using Samacheer Kalvi Business Mathematics and Statistics [English] Class 12 TN Board solutions Integral Calculus – 1 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 Samacheer Kalvi Solutions are essential questions that can be asked in the final exam. Maximum Tamil Nadu Board of Secondary Education Business Mathematics and Statistics [English] Class 12 TN Board students prefer Samacheer Kalvi Textbook Solutions to score more in exams.
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