Students can download 12th Business Maths Chapter 2 Integral Calculus I Additional Problems and Answers, Samacheer Kalvi 12th Business Maths Book Solutions Guide Pdf helps you to revise the complete Tamilnadu State Board New Syllabus and score more marks in your examinations.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems

I. One Mark Questions

Choose the correct answer.

Question 1.
The integral of \(\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)\) equals _____
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 1
Answer:
(c) \(\frac{2}{3} x^{\frac{3}{2}}+2 x^{\frac{1}{2}}+c\)

Question 2.
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 2
Answer:
(a) \(x^{4}+\frac{1}{x^{3}}-\frac{129}{8}\)
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 3

Question 3.
\(\int \frac{\sin ^{2} x-\cos ^{2} x}{\sin ^{2} x \cos ^{2} x} d x\) is equal to _____
(a) tan x + cot x + c
(b) tan x + cosec x + c
(c) -tan x + cot x + c
(d) tan x – sec x + c
Answer:
(a) tan x + cot x + c
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 4

Question 4.
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 5
Answer:
(d) \(\frac{\pi}{12}\)
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 6

Question 5.
The value of \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(x^{3}+x \cos x+\tan ^{5} x\right) d x\) is ______
(a) 0
(b) 2
(c) π
(d) 1
Answer:
(a) 0
Hint:
Let f(x) = x3 + x cos x + tan5 x
f(-x)= -x3 – x cos x – tan5 x = -f(x)
So f(x) is odd function.
Integral is 0.

Question 6.
Fill in the blanks.
(a) \(\int_{0}^{\frac{\pi}{2}} \cos ^{3} x d x\) is equal to _____
(b) \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{31} x d x\) is equal to _________
Answer:
(a) \(\frac{2}{3}\)
(b) 0
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 7

Question 7.
Match the following.
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 8
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 9
Answer:
(a) – (iii)
(b) – (iv)
(c) – (v)
(d) – (ii)
(e) – (i)

Question 8.
State True or False.
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 10
Answer:
(a) True
(b) False
(c) True
(d) False
(e) False
(f) True

Question 9.
Which of the following is not equal to ∫ tan x sec2 x dx?
(a) \(\frac{1}{2} \tan ^{2} x\)
(b) \(\frac{1}{2} \sec ^{2} x\)
(c) \(\frac{1}{2 \cos ^{2} x}\)
(d) None of these
Answer:
(d) None of these
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 11

Question 10.
∫ex (cos x – sin x) dx is equal to ______
(a) ex sin x + c
(b) ex cos x + c
(c) -ex cos x + c
(d) -ex sin x + c
Answer:
(b) ex cos x + c
Hint:
Let f(x) = cos x
f'(x) = -sin x
∫ex [f(x) + f'(x)] dx = ex f(x) + c

Question 11.
\(\int \frac{1-\cos 2 x}{1+\cos 2 x} d x\) is _____
(a) tan x – x + c
(b) x + tan x + c
(c) x – tan x + c
(d) -x – cot x + c
Answer:
(a) tan x – x + c
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 12

Question 12.
\(\int_{0}^{\frac{\pi}{2}} \cos x e^{\sin x} d x\) is equal to ______
(a) e = 1
(b) 1 – e
(c) \(e^{\frac{\pi}{2}}-1\)
(d) \(1-e^{\frac{\pi}{2}}\)
Answer:
(a) e = 1
Hint:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 13

Question 13.
Which of the following is an even function?
(a) sin x
(b) ex – e-x
(c) x cos x
(d) cos x
Answer:
(d) cos x

Question 14.
Which of the following is neither odd nor even function?
(a) x sin x
(b) x2
(c) e-x
(d) x cos x
Answer:
(c) e-x

Question 15.
∫sec2 (7 – 4x) dx equal to ______
(a) tan (7 – 4x)
(b) -tan (7 – 4x)
(c) –\(\frac{1}{4}\) tan (7 – 4x)
(d) \(\frac{1}{4}\) tan (7 – 4x)
Answer:
(c) – \(\frac{1}{4}\) tan (7 – 4x)

II. 2 Mark Questions.

Question 1.
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 14
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 15

Question 2.
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 16
Answer:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 17

Question 3.
\(\int\left(e^{x}+e^{-x}\right)^{2} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 18

Question 4.
Find \(\int x^{5} \sqrt{3+5 x^{6}} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 19

Question 5.
\(\int \frac{(x+1)(x+\log x)^{2}}{x} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 20
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 21

Question 6.
\(\int \frac{e^{2 x}-1}{e^{2 x}+1} d x\)
Solution:
Dividing numerator and denominator by ex, we get \(\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x\)
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 22

Question 7.
\(\int \frac{1}{x+\sqrt{x}} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 23

III. 3 and 5 Mark Questions.

Question 1.
Find \(\int \frac{e^{x}(x+1)}{(x+3)^{3}} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 24

Question 2.
\(\int \frac{1}{2 x^{2}-x-1} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 25
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 26

Question 3.
\(\int \frac{1}{1-3 \sin ^{2} x} d x\)
Solution:
Divide the numerator and denominator by cos2 x
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 27

Question 4.
\(\int \frac{d x}{e^{x}-e^{-x}}\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 28
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 29

Question 5.
Evaluate \(\int_{-1}^{2}(7 x-5) d x\) as the limit of a sum.
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 30

Question 6.
Evaluate \(\int_{1}^{2}\left(x^{2}-1\right) d x\) as the limit of a sum.
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 31
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 32

Question 7.
Evaluate \(\int_{1}^{2} \frac{1}{x\left(x^{4}+1\right)} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 33

Question 8.
Evaluate \(\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 34
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 35

Question 9.
Evaluate \(\int_{0}^{1} x(1-x)^{5} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 36

Question 10.
Evaluate \(\int \frac{2 x+3}{\sqrt{x^{2}+x+1}} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 37
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 38

Question 11.
∫(x2 + 1) log x dx
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 39

Question 12.
\(\int \sqrt{x^{2}+4 x-5} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 40

Question 13.
\(\int_{0}^{\frac{\pi}{4}}\left(2 \sec ^{2} x+x^{3}+2\right) d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 41
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 42

Question 14.
Evaluate \(\int_{0}^{1} \frac{2 x}{\left(x^{2}+1\right)\left(x^{2}+2\right)} d x\)
Solution:
Let x2 = t, then 2x dx = dt
when x = 0, t = 0 and x = 1, t = 1
so integral becomes, \(\int_{0}^{1} \frac{d t}{(t+1)(t+2)}\)
We use partial fractions to proceed further
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 43

Question 15.
\(\int_{0}^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\sin x \cos x} d x\)
Solution:
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 44
Samacheer Kalvi 12th Business Maths Solutions Chapter 2 Integral Calculus I Additional Problems 45

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