Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Students can Download Chapter 7 Integrals Questions and Answers, Plus Two Maths Chapter Wise Questions and Answers helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Plus Two Maths Integrals Three Mark Questions and Answers

Question 1.
Integrate the following. (3 Score each)

  1. ∫sin x sin 2x sin 3 xdx
  2. ∫sec2x cos22x dx

Answer:
1. We have sinxsin2xsin3x
= 1/2 (2sinxsin3x) sin2x
= 1/2 (cos2x – cos4x) sin2x
= 1/4 (2sin2xcos2x – 2cos4xsi n2x)
= 1/4 [sin4x – (sin6x – sin2x)]
= 1/4(sin4x + sin2x – sin6x)
∫sin x sin 2x sin 3 xdx
= \(\frac{1}{4}\) ∫(sin 4x + sin 2x – sin 6x)dx
= –\(\frac{1}{16}\) cos4x – \(\frac{1}{8}\) cos2x + \(\frac{1}{24}\) cos6x + c.

2. sec2x cos22x = \(\frac{\left(2 \cos ^{2} x-1\right)^{2}}{\cos ^{2} x}\)
= \(\left(\frac{2 \cos ^{2} x}{\cos x}-\frac{1}{\cos x}\right)^{2}\) = (2cosx – secx)2
= 4cos2x + sec2x – 4
= 2(1 + cos2x) + sec2x – 4
= 2cos2x + sec2x – 2
∫sec2 x cos2 2x dx = ∫(2 cos 2x + sec2 x – 2)dx
= sin 2x + tan x – 2x + c.

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 2.
Find \(\int \frac{2+\sin 2 x}{1+\cos 2 x} e^{x} d x\)?
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 1
= ∫ex [sec2 x + tan x]dx
= ∫ex[tanx + sec2x]dx = ex tanx + c.

Question 3.
Evaluate \(\int \frac{\sec ^{2} x d x}{\sqrt{\tan ^{2} x+4}}\)?
Answer:
Put tanx = u, sec2xdx = dy
Plus Two Maths Integrals 3 Mark Questions and Answers 2

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 4.
Find the following integrals.
Plus Two Maths Integrals 3 Mark Questions and Answers 3
Answer:
(i) I = \(\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1+\cos ^{2} x} d x\)
Put cosx = t ⇒ -sin xdx = dt
When x = 0 ⇒ t = cos0 = 1,
Plus Two Maths Integrals 3 Mark Questions and Answers 4

(ii) I = \(\int_{0}^{1} x e^{x^{2}} d x\)
Put x2 = t ⇒ 2xdx = dt
When x = 0 ⇒ t = 0,
x = 1 ⇒ t = 1
I = \(\frac{1}{2} \int_{0}^{1} e^{t} d t\) =
Plus Two Maths Integrals 3 Mark Questions and Answers 5
= [e1 – e0] = e – 1.
Plus Two Maths Integrals 3 Mark Questions and Answers 6
Put sin x = t ⇒ cos xdx = dt
When x = 0 ⇒ t = sin0 = 0,
Plus Two Maths Integrals 3 Mark Questions and Answers 7

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

(iv) I = \(\int_{0}^{2} x \sqrt{x+2} d x\)
Put x + 2 = t2 ⇒ dx = 2tdt
When x = 0 ⇒ t = \(\sqrt{2}\), x = 2 ⇒ t = 2
Plus Two Maths Integrals 3 Mark Questions and Answers 8

(v) I = \(\int_{0}^{\frac{\pi}{2}} \sqrt{\sin x} \cos x d x\)
Put sin x = t ⇒ cos xdx = dt
When x = 0 ⇒ t = sin0 = 0,
Plus Two Maths Integrals 3 Mark Questions and Answers 9
Plus Two Maths Integrals 3 Mark Questions and Answers 10
Put tan x = t ⇒ sec2 xdx = dt
When x = 0 ⇒ t = tan 0 = 0,
Plus Two Maths Integrals 3 Mark Questions and Answers 11

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 5.
(i) If f (x) is an odd function, then \(\int_{-a}^{a} f(x)\) = ?
(a) 0
(b) 1
(c) 2\(\int_{0}^{a} f(x)\) dx
(d) 2a
Evaluate
(ii) \(\int_{-\pi / 2}^{\pi / 2} \sin ^{99} x \cdot \cos ^{100} x d x\)
(iii) \(\int_{-1}^{1} e^{|x|} d x\)
Answer:
(i) (a) 0.

(ii) Here, f(x) = sin99x.cos100x .then,
f(-x) = sin99(- x).cos100(- x) = – sin99 x. cos100 x = -f(x)
∴ odd function ⇒ \(\int_{-\pi / 2}^{\pi / 2} \sin ^{99} x \cdot \cos ^{100} x d x=0\).

(iii) Here, f(x) = e|x|, f(-x) = e|-x| = e|x| = f(x)
∴ even function.
Plus Two Maths Integrals 3 Mark Questions and Answers 12
we have |x| = x, 0 ≤ x ≤ 1
Plus Two Maths Integrals 3 Mark Questions and Answers 13

Question 6.

  1. Show that cos2 x is an even function. (1)
  2. Evaluate \(\int_{-\pi / 4}^{\pi / 4} \cos ^{2} x d x\) (2)

Answer:
1. Let f(x) = cos2x ⇒ f(-x) = cos2 (-x) = cos2 x = f(x) even.

2.
Plus Two Maths Integrals 3 Mark Questions and Answers 14

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 7.
Find the following integrals.
Plus Two Maths Integrals 3 Mark Questions and Answers 15
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 16
Plus Two Maths Integrals 3 Mark Questions and Answers 17

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 8.
Find the following integrals.
Plus Two Maths Integrals 3 Mark Questions and Answers 18
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 19
Plus Two Maths Integrals 3 Mark Questions and Answers 20

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals
Add (1) and (2)
Plus Two Maths Integrals 3 Mark Questions and Answers 21
Plus Two Maths Integrals 3 Mark Questions and Answers 22

Plus Two Maths Integrals 3 Mark Questions and Answers 23
Plus Two Maths Integrals 3 Mark Questions and Answers 24

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 9.
Find the following integrals.

  1. \(\int \frac{1}{3+\cos x} d x\)
  2. \(\int \frac{2 x}{x^{2}+3 x+2} d x\)

Answer:
1. \(\int \frac{1}{3+\cos x} d x\)
Put t = tanx/2 ⇒ dt = 1/2 sec2 x/2 dx
Plus Two Maths Integrals 3 Mark Questions and Answers 25

2. \(\int \frac{2 x}{x^{2}+3 x+2} d x\) = \(\int \frac{2 x}{(x+2)(x+1)} d x\)
Plus Two Maths Integrals 3 Mark Questions and Answers 26
2x = A(x + 1) + B (x + 2)
when x = -1, -2 = B ; B = -2
when x = -2, -4 = -A ; A = 4
Plus Two Maths Integrals 3 Mark Questions and Answers 27
= 4log(x + 2) – 2log (x + 1) + C.

Plus Two Maths Integrals Four Mark Questions and Answers

Question 1.
Find the following integrals.
Plus Two Maths Integrals 3 Mark Questions and Answers 28
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 29
x2 + x +1 = A(x2 + 1) + (Bx + C)(x + 2)
Put x = -2 ⇒ 4 – 2 + 1 = 5A ⇒ A = \(\frac{3}{5}\)
Equating the coefficients of x2
⇒ 1 = A + B ⇒ B = 1 – \(\frac{3}{5}\) = \(\frac{2}{5}\)
Equating the constants
⇒ 1 = A + 2C ⇒ 2C = 1 – \(\frac{3}{5}\) = \(\frac{2}{5}\) ⇒ C = \(\frac{1}{5}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 30
Plus Two Maths Integrals 3 Mark Questions and Answers 31
⇒ 1 = A(x – 1) + B(x + 3)
Put x = 1 ⇒ 1 = 2A ⇒ A = \(\frac{1}{2}\)
Put x = -3 ⇒ 1 = -4B ⇒ B = – \(\frac{1}{4}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 32
Plus Two Maths Integrals 3 Mark Questions and Answers 33
Equating the constants; ⇒ 1 = A
Equating the coefficients if t;
⇒ 0 = A + B ⇒ B = -1
Plus Two Maths Integrals 3 Mark Questions and Answers 34

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 2.
Find the following integrals.

  1. ∫ e2x sin3xdx
  2. ∫ x sin-1xdx

Answer:
1. I = ∫e2x sin3xdx = ∫ sin 3x × e2xdx
Plus Two Maths Integrals 3 Mark Questions and Answers 35
Plus Two Maths Integrals 3 Mark Questions and Answers 36

2. ∫ x sin-1xdx = ∫ sin-1x × xdx
Plus Two Maths Integrals 3 Mark Questions and Answers 37

Question 3.
(i) Which of the following is the value of \(\int \frac{d x}{\sqrt{a^{2}-x^{2}}}\)? (1)
Plus Two Maths Integrals 3 Mark Questions and Answers 38
(ii) Evaluate \(\int \frac{2 x}{x^{2}+3 x+2} d x\) (3)
Answer:
(i) [sin-1\(\frac{x}{a}\) + c]

(ii)
Plus Two Maths Integrals 3 Mark Questions and Answers 39
⇒ 2x = A(x + 1) + B(x + 2) ⇒
Put x = -2 and x = -1, we get A = 4, B = -2
Plus Two Maths Integrals 3 Mark Questions and Answers 40

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 4.

  1. Choose the correct answer from the bracket.
    ∫ex dx = — (e2x + c, e-x + c, e2x + c) (1)
  2. Evaluate: ∫ ex sinxdx

Answer:
1. ex + c

2. I = ∫ex sinxdx = sinx.ex – ∫cos x.exdx
= sin x.ex – (cos x.ex – ∫(- sin x).ex dx)
= sinx.ex – cosxex – ∫sinx.exdx
= sin x.ex – cos xex – I
2I = sin x.ex – cos xex
I = \(\frac{1}{2}\)ex(sinx – cosx) + c.

Question 5.
(i) f(x)∫g(x) dx – ∫(f'(x)∫g(x) dx)dx (1)
(a) ∫f'(x)g{x)dx
(b) ∫f(x)g'(x)dx
(c) ∫\(\frac{f(x)}{g(x)}\)dx
(d) ∫f(x)g(x)dx
(ii) Integrate sin-1\(\sqrt{\frac{x}{a+x}}\)dx w.r.to x. (3)
Answer:
(i) (d) ∫f(x)g(x)dx

(ii) ∫sin-1\(\sqrt{\frac{x}{a+x}}\)dx,
Put x = a tan2θ, θ = tan-1\(\sqrt{\frac{x}{a}}\)
⇒ dx = 2a tanθ sec2θ dθ
I = ∫sin-1\(\left(\frac{\tan \theta}{\sec \theta}\right)\) 2a tanθ sec2θ dθ
= ∫sin-1(sinθ)2a tanθ sec2θ dθ
= 2a∫θ tanθ sec2θ dθ
Put tanθ = t, θ = tan-1 t ⇒ sec2θ dθ = dt
= 2a ∫ tan-1 t (t) dθ
Plus Two Maths Integrals 3 Mark Questions and Answers 41
= a[tan2θ.θ – tanθ + θ] + c
= a[θ(1 + tan2θ) – tanθ] + c
Plus Two Maths Integrals 3 Mark Questions and Answers 42

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 6.
Match the following. (4)
Plus Two Maths Integrals 3 Mark Questions and Answers 43
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 44

Question 7.
Evaluate \(\int \frac{x}{\sqrt{x+a}+\sqrt{x+b}} d x\)?
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 45

Question 8.
Match the following.
Plus Two Maths Integrals 3 Mark Questions and Answers 46
Answer:
1.
Plus Two Maths Integrals 3 Mark Questions and Answers 47

2. ∫sec x(sec x + tan x)dx = ∫(sec2 x + sec x. tan x)dx
= tanx + secx + c.

3. ∫e3xdx = \(\frac{e^{3 x}}{3}\) + c.

4. ∫(sin x + cos x)dx = sin x – cosx + c.

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 9.
Consider the integral I = \(\int \frac{x \sin ^{-1} x}{\sqrt{1-x^{2}}} d x\)?

  1. What substitution can be given for simplifying the above integral? (1)
  2. Express I in terms of the above substitution. (1)
  3. Evaluate I. (2)

Answer:
1. Substitute sin-1 x = t.

2. We have, sin-1 x = t ⇒ x = sint
Differentiating w.r.t. x; we get,
\(\frac{1}{\sqrt{1-x^{2}}}\)dx = dt
∴ I = ∫t sin t dt.

3. I = ∫t sin t dt = t.(-cost) -∫(-cost)dt = -t cost + sint + c
= -sin-1 x. cos (sin-1 x) + sin(sin-1 x) + c
x – sin-1 x.cos(sin-1 x) + c.

Question 10.
Evaluate \(\int_{0}^{\pi / 4} \log (\tan x) d x\).
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 48

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 11.
Find the following integrals.

  1. \(\int \frac{\sec ^{2} x}{\cos e c^{2} x} d x\) (2)
  2. \(\int \frac{1}{x^{2}-6 x+13} d x\) (2)

Answer:
1. \(\int \frac{\sec ^{2} x}{\cos e c^{2} x} d x\) = \(\int \frac{\sin ^{2} x}{\cos ^{2} x} d x\) = ∫tan2 xdx
= ∫(sec2x – 1)dx = tanx – x + c.

2. \(\int \frac{1}{x^{2}-6 x+13} d x\)
Plus Two Maths Integrals 3 Mark Questions and Answers 49

Question 12.
Match the following. Justify your answer.
Plus Two Maths Integrals 3 Mark Questions and Answers 50
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 51

Question 13.
(i) ∫sin2x dx = ? (1)
(a) 2 cos x + c
(b) -2 sin x + c
(c) \(\frac{\cos 2 x}{2}\) + c
(d) \(-\frac{\cos 2 x}{2}\) + c
(ii) Evaluate ∫ex sin 2x dx (3)
Answer:
(i) (d) \(-\frac{\cos 2 x}{2}\) + c.

(ii) Consider I = ∫ex sin 2x dx
= ∫sin 2x. exdx = sinx.ex – 2∫cos 2x. exdx
= sin 2x.ex – 2 (cos 2x.ex + 2∫sin 2x. exdx)
= sin 2x. ex – 2 cos 2x ex – 4 ∫sin 2x. exdx
= sin 2x. ex – 2 cos 2x ex – 4I
5 I = sin 2x. ex – 2 cos 2x ex
I = \(\frac{e^{x}}{5}\) (sin 2x – 2 cos 2x).

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 14.

  1. Resolve \(\frac{x^{2}+1}{x^{2}-5 x+6}\) into partial fractions. (2)
  2. Hence evaluate ∫\(\frac{x^{2}+1}{x^{2}-5 x+6}\). (2)

Answer:
1.
Plus Two Maths Integrals 3 Mark Questions and Answers 52

2.
Plus Two Maths Integrals 3 Mark Questions and Answers 53
5x – 5 = A(x – 2) + B(x – 3)
x = 2, 5 = -B, B = -5
x = 3, 10 = A, A = 10
(1) ⇒ I = ∫ 1dx + ∫\(\frac{10}{x-3}\) dx – ∫\(\frac{5}{x-2}\) dx
= x + 10log(x – 3) – 5log(x – 2) + c.

Question 15.
Evaluate \(\int_{0}^{4}\) xdx as a limit of sum.
Answer:
By definition,
\(\int_{a}^{b}\) f(x) dx =
(b – a)\(\lim _{n \rightarrow \infty} \frac{1}{n}\){f(a) + f(a + h) +…….+f(a + {n – 1)h)}
Here, a = 0, b = 4, f(x) = x, h = \(\frac{4-0}{n}=\frac{4}{n}\) ⇒ nh = 4
Plus Two Maths Integrals 3 Mark Questions and Answers 54

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 16.

  1. Define the real valued function f(x) = |x2 + 2x – 3| (2)
  2. Evaluate \(\int_{0}^{2}\)|x2 + 2x – 3|dx. (2)

Answer:
1. f(x) = |x2 + 2x – 3| = |(x – 1) (x + 3)|
We have;
Plus Two Maths Integrals 3 Mark Questions and Answers 55

2. I = \(\int_{0}^{2}\)|x2 + 2x – 3|dx
Plus Two Maths Integrals 3 Mark Questions and Answers 56

Question 17.
Consider the function f(x) = |x|+|x + 1|

  1. Define the function f (x) in the interval [-2, 1]. (2)
  2. Find the integral \(\int_{-2}^{1}\) f(x) dx (2)

Answer:
1. Given, f(x) = |x|+|x + 1|.
We have,
Plus Two Maths Integrals 3 Mark Questions and Answers 57
Combining these two functions, we get the function f(x).
Plus Two Maths Integrals 3 Mark Questions and Answers 58

2.
Plus Two Maths Integrals 3 Mark Questions and Answers 59

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 18.
Evaluate \(\int_{\sqrt{6}}^{\sqrt{3}} \frac{d x}{1+\sqrt{\tan x}} d x\). (4)
Answer:
Plus Two Maths Integrals 3 Mark Questions and Answers 60

Plus Two Maths Integrals Six Mark Questions and Answers

Question 1.
(i) Fill in the blanks. (3)
(a) ∫ tan xdx = —
(b) ∫ cos xdx = —
(c) ∫\(\frac{1}{x}\)dx = —
(ii) Evaluate ∫sin3 xcos2 xdx (3)
Answer:
(i) (a) log|secx| + c
(b) sinx + c
(c) log|x| + c.

(ii) ∫sin3 xcos2 xdx = ∫sin2 xcos2 x sin xdx
= ∫(1 – cos2 x)cos2 x sin xdx
Put cos x = t ⇒ – sin xdx = dt
∴ ∫(1 – cos2 x)cos2 xsin xdx = -∫(1 – t2 )t2dt
= ∫(t4 – t2)dt = \(\frac{t^{5}}{5}-\frac{t^{3}}{3}\) + c
= \(\frac{\cos ^{5} x}{5}-\frac{\cos ^{3} x}{3}\) + c.

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 2.
Find the following integrals.
Plus Two Maths Integrals 3 Mark Questions and Answers 61
Answer:
(i) I = ∫(3x – 2)\(\sqrt{x^{2}+x+1} d x\)
Let 3x – 2 = A(2x + 1) + B
⇒ 3 = 2 A ⇒ A = \(\frac{3}{2}\)
⇒ -2 = A + B ⇒ -2 = \(\frac{3}{2}\) + B
⇒ B = -2 – \(\frac{3}{2}\) = – \(\frac{7}{2}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 62
Plus Two Maths Integrals 3 Mark Questions and Answers 63
Using (2) and (3) in (1) we have;
Plus Two Maths Integrals 3 Mark Questions and Answers 64

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

(ii) I = \(\int \frac{2 x-3}{x^{2}+3 x-18} d x\)
Let 2x – 3 = A(2x + 3) + B
⇒ 2 = 2A ⇒ A = 1
⇒ -3 = 3A + B ⇒ -3 = 3 + B ⇒ B = -6
Plus Two Maths Integrals 3 Mark Questions and Answers 65
Plus Two Maths Integrals 3 Mark Questions and Answers 66

(iii) I = \(\int \frac{5 x+2}{1+2 x+3 x^{2}} d x\)
Let 5x + 2 = A{6x + 2) + B
⇒ 5 = 6 A ⇒ A = \(\frac{5}{6}\)
⇒ 2 = 2A + B ⇒ 2 = \(\frac{5}{3}\) + B ⇒ 2 – \(\frac{5}{3}\) = \(\frac{1}{3}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 67
Plus Two Maths Integrals 3 Mark Questions and Answers 68

(iv) I = \(\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x\)
Let 5x + 3 = A(2x + 4) + B
⇒ 5 = 2A ⇒ A = \(\frac{5}{2}\)
⇒ 3 = 4A + B ⇒ 3 = 10 + B ⇒ B = -7
Plus Two Maths Integrals 3 Mark Questions and Answers 69
Plus Two Maths Integrals 3 Mark Questions and Answers 70
Using (2) and (3) in (1) we have;
Plus Two Maths Integrals 3 Mark Questions and Answers 71

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 3.
Consider the expression \(\frac{1}{x^{3}-1}\)

  1. Split it into partial fraction. (2)
  2. Evaluate ∫ \(\frac{1}{x^{3}-1}\) dx (4)

Answer:
1.
Plus Two Maths Integrals 3 Mark Questions and Answers 72
1 = A (x2 + x + 1) + (Bx + c)(x + 1),
Put x = -1 ⇒ 1 = A(1 + 1 + 1) ⇒ A= \(\frac{1}{3}\)
Equating like terms.
0 = A + B ⇒ B = – \(\frac{1}{3}\), 1 = A + C ⇒ C = \(\frac{2}{3}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 73

2.
Plus Two Maths Integrals 3 Mark Questions and Answers 74
Put, x – 2 = D (2x – 1) + E ,
1 = 2 D ⇒ D = \(\frac{1}{2}\),
-2 = -D + E ⇒ E = –\(\frac{3}{2}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 75
Plus Two Maths Integrals 3 Mark Questions and Answers 76

Question 4.
(i) Match the following (4)
Plus Two Maths Integrals 3 Mark Questions and Answers 77
(ii) Consider the function f(x) = \(\frac{x^{4}}{x+1}\) Evaluate ∫f(x)dx (2)
Answer:
(i)
Plus Two Maths Integrals 3 Mark Questions and Answers 78

(ii) Here the numerator is of degree 4 and denominator of degree 1. So to make it a proper fraction we have to divide Nr by Dr.
Plus Two Maths Integrals 3 Mark Questions and Answers 79

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 5.

  1. Evaluate the as \(\int_{0}^{2}\)x2dx the limit of a sum. (3)
  2. Hence evaluate \(\int_{-2}^{2}\)x2dx (1)
  3. If \(\int_{0}^{2}\) f(x)dx = 5 and \(\int_{-2}^{2}\) f(x)dx = 0, then \(\int_{-2}^{0}\) f(x)dx = …….. (2)

Answer:
1. Here the function is f(x) = x2, a = 0, b = 2 and h = \(\frac{b-a}{n}=\frac{2}{n}\)
\(\int_{0}^{2}\)x2dx =
Plus Two Maths Integrals 3 Mark Questions and Answers 80

2. \(\int_{-2}^{2}\) x2dx = 2 \(\int_{0}^{2}\)x2dx = \(\frac{16}{3}\)

3.
Plus Two Maths Integrals 3 Mark Questions and Answers 81

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 6.
Find ∫\(\sqrt{\tan x}\)xdx.
Answer:
Given;
I = ∫\(\sqrt{\tan x}\)xdx,
Put tanx = t2 ⇒ sec2xdx = 2tdt ⇒ dx = \(\frac{2 t d t}{1+t^{4}}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 82
Plus Two Maths Integrals 3 Mark Questions and Answers 83
Plus Two Maths Integrals 3 Mark Questions and Answers 84

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 7.
(i) Match the following. (2)
Plus Two Maths Integrals 3 Mark Questions and Answers 85
(ii) Integrate \(\frac{\sec ^{2} x}{5 \tan ^{2} x-12 \tan x+14}\) w.r.to x. (4)
Answer:
(i)
Plus Two Maths Integrals 3 Mark Questions and Answers 86
Plus Two Maths Integrals 3 Mark Questions and Answers 87
Plus Two Maths Integrals 3 Mark Questions and Answers 88

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 8.

  1. Evaluate \(\int_{0}^{1} \sqrt{x} d x\) (1)
  2. If \(\int_{0}^{a} \sqrt{x} d x=2 a \int_{0}^{\pi / 2} \sin ^{3} x d x\), find the value of a. (3)
  3. Hence find \(\int_{a}^{a+1}\)x dx. (2)

Answer:
1.
Plus Two Maths Integrals 3 Mark Questions and Answers 89

2. Given;
Plus Two Maths Integrals 3 Mark Questions and Answers 90

3. When a = 0
Plus Two Maths Integrals 3 Mark Questions and Answers 91
When, a = 4
Plus Two Maths Integrals 3 Mark Questions and Answers 92

Question 9.
(i) Let f (x) be a function, then \(\int_{0}^{a}\) f(x) dx = ? (1)
(a) 2 \(\int_{0}^{a}\) f(x – a) dx
(b) \(\int_{0}^{a}\) f(a – x) dx
(c) f(a)
(d) 2\(\int_{0}^{a}\) f(a – x) dx
Evaluate
Plus Two Maths Integrals 3 Mark Questions and Answers 93
Answer:
(i) (b) \(\int_{0}^{a}\) f(a – x) dx

(ii)
Plus Two Maths Integrals 3 Mark Questions and Answers 94
(1) + (2)
Plus Two Maths Integrals 3 Mark Questions and Answers 95
⇒ I = 1.

(iii)
Plus Two Maths Integrals 3 Mark Questions and Answers 96

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 10.
Find the following integrals.

  1. ∫\(\frac{2 e^{x}}{e^{3 x}-6 e^{2 x}+11 e^{x}-6} d x\)
  2. ∫\(\frac{(3 \sin x-2) \cos x}{5-\cos ^{2} x-4 \sin x} d x\)

Answer:
1.
Plus Two Maths Integrals 3 Mark Questions and Answers 97
⇒ 1 = A(t – 2)(t – 3) + B(t – 1)(t – 3) + C(t – 1)(t – 2)
Put t = 1 ⇒ 1 = A(-1)(-2) ⇒ A = \(\frac{1}{2}\)
Put t = 2 ⇒ 1 = B(1)(-1) ⇒ B = -1
Put t = 3 ⇒ 1 = B(2)(1) ⇒ B = \(\frac{1}{2}\)
Plus Two Maths Integrals 3 Mark Questions and Answers 98
Plus Two Maths Integrals 3 Mark Questions and Answers 99

2. I = ∫\(\frac{(3 \sin x-2) \cos x}{5-\cos ^{2} x-4 \sin x} d x\)dx
Put sin x = t ⇒ cosxdx = dt
Plus Two Maths Integrals 3 Mark Questions and Answers 100
⇒ 3t – 2 = A(t – 2) + B
Equating the coefficients if t; ⇒ 3 = A
Equating the constants
⇒ -2 = -2A + B ⇒ -2 = -6 + B ⇒ B = 4
Plus Two Maths Integrals 3 Mark Questions and Answers 101

Plus Two Maths Chapter Wise Questions and Answers Chapter 7 Integrals

Question 11.

  1. Find ∫\(\frac{1}{x^{2}+a^{2}}\)dx (1)
  2. Show that 3x + 1 = \(\frac{3}{4}\)(4x – 2) + \(\frac{5}{2}\) (2)
  3. Evaluate \(\int \frac{3 x+1}{2 x^{2}-2 x+3} d x\) (3)

Answer:
1. ∫\(\frac{1}{x^{2}+a^{2}}\)dx = 1/a tan-1 x/a + c.

2. 3x + 1 = A \(\frac{d}{d x}\)(2x2 – 2x + 3) + B
= A(4x – 2) + B
3 = 4A; A = 3/4
1 = -2A + B
1 = -3/2 + B, B = 1 + 3/2 = 5/2
∴ 3x + 1 = 3/4(4x – 2) + 5/2

3.
Plus Two Maths Integrals 3 Mark Questions and Answers 102

Leave a Comment