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Integral Calculus I Exercise 2.12 – 12th Text Book Solutions (Business Maths)

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Dear Samacheer Kalvi Students, here are the answers for book back exercise 2.12 in Business Maths Chapter 2 Integral Calculus I. If you have any doubts, please reach out to us in the comments section.

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Important Formulas in Integral Calculus I

Text Book Solutions for Integral Calculus I Exercise 2.1

Text Book Solutions for Integral Calculus I Exercise 2.2

Text Book Solutions for Integral Calculus I Exercise 2.3

Text Book Solutions for Integral Calculus I Exercise 2.4

Text Book Solutions for Integral Calculus I Exercise 2.5

Text Book Solutions for Integral Calculus I Exercise 2.6

Text Book Solutions for Integral Calculus I Exercise 2.7

Text Book Solutions for Integral Calculus I Exercise 2.8

Text Book Solutions for Integral Calculus I Exercise 2.9

Text Book Solutions for Integral Calculus I Exercise 2.10

Text Book Solutions for Integral Calculus I Exercise 2.11

Exercise 2.11

Choose the correct answer.

1. \displaystyle\int \frac{1}{x^{3}}\: dx
a)\dfrac{-3}{x^{2}}+c
b.)\dfrac{-1}{2x^{2}}+c (ANSWER)
c.)\dfrac{-1}{3x^{2}}+c
d.)\dfrac{-2}{x^{2}}+c
2.\displaystyle \int2^{x}\: dx
a.\: 2^{x}\:log 2 +c
b.\: 2^{x}+c
c.\: \dfrac{2^{x}}{log \:2}+c (ANSWER)
d.\: \dfrac{log \:2 }{2^{x}}+c
3.\displaystyle \int \dfrac{sin2x}{2sinx}\: dx
a.\: sin x\:+c (ANSWER)
b.\dfrac{1}{2}\: sin x\:+c
c.\: cos\:x+c
d.\: \dfrac{1}{2}\: cos\:x+c
4.\displaystyle \int \dfrac{sin5x-sinx}{cos3x}\: dx
a.\: - cos 2x\:+c (ANSWER)
b.\:- cos 2x\:+c
c.\:\dfrac{-1}{4}\: cos\:2x+c
d.\: -4\: cos\:2x+c
5.\displaystyle \int \dfrac{log x}{x}\: dx, \:x> 0\: is
a.\: \dfrac{1}{2}\: (logx)^{2} +c (ANSWER)
b.\:\dfrac{-1}{2}\: (logx)^{2} +c
c.\:\dfrac{2}{x^{2}}+c
d.\: \dfrac{-2}{x^{2}}+c
6.\displaystyle \int \dfrac{e^{x}}{\sqrt{1+e^{x}}}\: dx \: is
a.\:\dfrac{e^{x}}{\sqrt{1+e^{x}}} +c
b.\:2\sqrt{1+e^{x}} +c (ANSWER)
c.\:\sqrt{1+e^{x}} +c
d.\: e^{e}\sqrt{1+e^{x}} +c
7.\displaystyle \int \sqrt{e^{x}}\: dx \: is
a.\:\sqrt{e^{x}} +c
b.\:2\sqrt{e^{x}} +c (ANSWER)
c.\:\dfrac{1}{2} \sqrt{e^{x}} +c
d.\: \dfrac{1}{2\sqrt{e^{x}}} +c
8.\displaystyle \int e^{2x}\left [ 2x^{2}+2x \right ] \: dx \: is
a.e^{2x}x^{2} +c (ANSWER)
b.\:x e^{2x} +c
c.\:2 x^{2}e^{2} +c
d.\: \dfrac{x^{2}e^{x}}{2} +c
9.\displaystyle \int \dfrac{e^{x}}{e^{x}+1} \: dx \: is
a.log \left |\dfrac{e^{x}}{e^{x}+1} \right |+c
b.\:log \left |\dfrac{e^{x}+1}{e^{x}} \right | +c
c.\:log \left |e^{x} \right |+c
d.\: log \left |e^{x}+1 \right | +c (ANSWER)
10.\displaystyle \int \left [\dfrac{9}{x-3}- \dfrac{1}{x+1} \right ] \: dx \: is
a.log \left |(x-3) \right |-log \left |(x+1) \right |+c
b.\:log \left |(x-3) \right |+log \left |(x+1) \right | +c
c.\:9log \left |(x-3) \right |-log \left |(x+1) \right |+c (ANSWER)
d.\: 9log \left |(x-3) \right |+log \left |(x+1) \right | +c
11.\displaystyle \int \dfrac{2x^{3}}{4+x^{4}} \: dx \: is
a.log \left |4+x^{4} \right |+c
b.\:\dfrac{1}{2} log \left |4+x^{4} \right | +c (ANSWER)
c.\:\dfrac{1}{4} log \left |4+x^{4} \right | +c
d.\: log \left |\dfrac{2x^{3}}{4+x^{4}} \right | +c
11.\displaystyle \int \dfrac{2x^{3}}{4+x^{4}} \: dx \: is
a.log \left |4+x^{4} \right |+c
b.\:\dfrac{1}{2} log \left |4+x^{4} \right | +c (ANSWER)
c.\:\dfrac{1}{4} log \left |4+x^{4} \right | +c
d.\: log \left |\dfrac{2x^{3}}{4+x^{4}} \right | +c
12.\displaystyle \int \dfrac{dx}{\sqrt{x^{2}-36}} \: is
a.\sqrt{x^{2}-36}+c
b.\:log \left |x+\sqrt{x^{2}-36} \right | +c(ANSWER)
c.\:\:log \left |x-\sqrt{x^{2}-36} \right | +c
d.\: \:log \left |x^{2}+\sqrt{x^{2}-36} \right | +c
13.\displaystyle \int \dfrac{2x+3}{\sqrt{x^{2}+3x+2}} \: dx \: is
a.\sqrt{x^{2}+3x+2} +c
b.\:2 \sqrt{x^{2}+3x+2} +c(ANSWER)
c.\:\:log \left (x^{2}+3x+2 \right ) +c
d.\: \dfrac{2}{3}\left (x^{2}+3x+2 \right ) +c
14.\displaystyle \int_{_{0}}^{1} ( 2x+1 )\: dx \: is
a. 1
b.2 (ANSWER)
c. 3
d. 4
15.\displaystyle \int_{_{2}}^{4} \dfrac{dx}{x} \: is
a. log 4
b. 0
c. log 2 (ANSWER)
d. log 8
16.\displaystyle \int_{_{0}}^{\infty} e^{-2x} \:dx\: is
a. 0
b. 1
c. 2
d.\dfrac{1}{2} (ANSWER)
17.\displaystyle \int_{_{-1}}^{1} x^{3}e^{x^{4}} \:dx\: is
a. 1
b.\:\displaystyle \int_{_{0}}^{1} x^{3}e^{x^{4}} \: dx
c. 0 (ANSWER)
d.\: e^{x^{4}}
18. If \: f(x) \:is \: a \: continuous \: function \: and \:a< c< b \: then \: \displaystyle \int_{a}^{c}\: f(x)\: dx + \displaystyle \int_{c}^{b}f(x)\: dx\: is
a. \displaystyle \int_{a}^{b} f(x) \: dx-\displaystyle \int_{a}^{c} f(x) \: dx
b. \displaystyle \int_{a}^{c} f(x) \: dx- \displaystyle \int_{a}^{b} f(x) \: dx
c.\displaystyle \int_{a}^{b} f(x) \: dx (ANSWER)
d. 0
19. \:The \: value \:of \: \displaystyle \int_{_{\frac{-\pi}{2}}}^{\frac{\pi}{2}}cos\: x\: dx\: is
a. 0
b. 2 (ANSWER)
c. 1
d. 4
20. \displaystyle \int_{{0}}^{1}\sqrt{x^{4}(1-x)^{2}}\: dx\: is
a.\frac{1}{12} (ANSWER)
b.\frac{-7}{12}
c.\frac{7}{12}
d.\frac{-1}{12}
21. If\: \displaystyle \int_{{0}}^{1}f(x)\: dx=1, \:\int_{{0}}^{1}x\:f(x)\: dx=a \: and \: \int_{{0}}^{1}x^{2}\:f(x)\: dx\:=a^{2},\: then \: \:\int_{{0}}^{1}(a-x)^{2}\:f(x)\: dx
a.4a^{2}
b. 0 (ANSWER)
c.2a^{2}
d. 1
22. \: The\: value\: of \: \displaystyle \int_{{2}}^{3}f(5-x)\: dx -\displaystyle \int_{{2}}^{3}f(x)\: dx\: is
a. 1
b. 0 (ANSWER)
c. -1
d. 5
23. \displaystyle \int_{{0}}^{4}\left (\sqrt{x} +\frac{1}{\sqrt{x}} \right )\: dx\: is
a.\dfrac{20}{3}
b.\dfrac{21}{3}
c.\dfrac{28}{3} (ANSWER)
d.\dfrac{1}{3}
24. \displaystyle \int_{{0}}^{\frac{\pi}{3}}tan x\: dx\: is
a. log 2 (ANSWER)
b. 0
c.log \sqrt{2}
d. 2 log 2
25. Using the factorial representation of the gamma function, which of the following is the solution for the gamma function \Gamma (n) dx\: is when n=8
a. 5040 (ANSWER)
b. 5400
c. 4500
d. 5540
26. \Gamma (n) dx\: is
a.(n-1)! (ANSWER)
b.(n)!
c. n\Gamma (n)
d.(n-1)\Gamma (n)
27. \Gamma (1) dx\: is
a. 0
b. 1 (ANSWER)
c. n
d. n!
28. \:If\: n> 0 \:then \:\Gamma (n) is
a. \displaystyle \int_{{0}}^{1} e^{-x} x^{n-1} \: dx
b.\displaystyle \int_{{0}}^{1} e^{-x} x^{n} \: dx
c.\displaystyle \int_{{0}}^{\infty} e^{x} x^{-n} \: dx
d.\displaystyle \int_{{0}}^{\infty} e^{-x} x^{n-1} \: dx (ANSWER)
29.\:\Gamma (\frac{3}{2}) is
a.\sqrt{\pi}
b.\dfrac{\sqrt{\pi}}{2} (ANSWER)
c.2\sqrt{\pi}
d.\dfrac{3}{2}
30.\:\displaystyle \int_{{0}}^{\infty}x^{4}e^{-x} \:dx\: is
a. 12
b. 4
c. 4! (ANSWER)
d. 64

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