Exercise 13C page: 678
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Transcript of Exercise 13C page: 678
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
Exercise 13C page: 678 1.
Solution:
It is given that
∫ x ex dx
We can write it as
By integrating w.r.t x
= x ex – ∫ 1 ex dx
So we get
= x ex – ex + c
By taking ex as common
= ex (x – 1) + c
2.
Solution:
It is given that
∫ x cos x dx
We can write it as
By integrating w.r.t x
= x sin x – ∫ 1 sin x dx
So we get
= x sin x + cos x + c
3.
Solution:
It is given that
∫ x e2x dx
We can write it as
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
4.
Solution:
It is given that
∫ x sin 3x dx
We can write it as
5.
Solution:
It is given that
∫ x cos 2x dx
We can write it as
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
So we get
6.
Solution:
It is given that
∫ x log 2x dx
We can write it as
7.
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
Solution:
It is given that
∫ x cosec2 x dx
We can write it as
By integrating w.r.t x
= x (- cot x) – ∫ 1 (- cot x) dx
So we get
= – x cot x + ∫ cot x dx
Again by integration we get
= – x cot x + log |sin x| + c
8.
Solution:
It is given that
∫ x2 cos x dx
We can write it as
By integrating w.r.t x
= x2 sin x – ∫ 2x sin x dx
So we get
= x2 sin x – 2 [∫ x sin x dx]
Now apply by the part method
= x2 sin x – 2 ∫ x sin x dx
Again by integration
We get
= x2 sin x – 2 [x (- cos x) – ∫ 1. (- cos x) dx]
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
By integrating w.r.t x
= x2 sin x – 2 (- x cos x + sin x) + c
On further simplification
= x2 sin x + 2x cos x – 2 sin x + c
9.
Solution:
It is given that
∫ x sin2 x dx
We can write it as
sin2 x = (1 + cos 2x)/ 2
By integration
It can be written as
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
10.
Solution:
It is given that
∫ x tan2 x dx
We can write it as
tan2 x = sec2 x – 1
So we get
∫ x tan2 x dx = ∫ x ( sec2 x – 1) dx
By further simplification
= ∫ x sec2 x dx – ∫ x dx
Taking first function as x and second function as sec2 x
∫ x sec2 x dx – ∫ x dx
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
We get
By integration w.r.t x
= {x tan x – ∫ tan x dx} – x2/ 2
It can be written as
= x tan x – log |sec x| – x2/ 2 + c
So we get
= x tan x – log |1/cos x| – x2/ 2 + c
Here
= x tan x + log |cos x| – x2/ 2 + c
11.
Solution:
It is given that
∫ x2 ex dx
By integrating w.r.t x
We get
= x2 ex – 2 [x ex – ∫1. ex dx]
By further simplification
= x2 ex – 2 [x ex – ex] + c
By multiplication
= x2 ex – 2x ex + 2 ex + c
By taking ex as common
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
= ex (x2 – 2x + 2) + c
13.
Solution:
It is given that
∫ x2 e3x dx
We can write it as
14.
Solution:
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
It is given that
∫ x2 sin2 x dx
We know that
Now by integrating the second part we get
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
15.
Solution:
It is given that
∫ x3 log 2x dx
We can write it as
16.
Solution:
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
It is given that
17.
Solution:
It is given that
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
18.
Solution:
It is given that
We can write it as
= t et – ∫1. et dt
So we get
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
= t et – et + C
Now by replacing t with x2
19.
Solution:
It is given that
∫ x sin3 x dx
We know that
sin 3x = 3 sin x – 4 sin3 x
It can be written as
sin3 x = (3 sin x – sin 3x)/4
We get
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
So we get
20.
Solution:
It is given that
∫ x cos3 x dx
We know that
cos3 x = (cos 3x + 3 cos x)/4
It can be written as
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
21.
Solution:
It is given that
∫ x3 cos x2 dx
We can write it as
∫ x x2 cos x2 dx
Take x2 = t
So we get 2x dx = dt
x dx = dt/2
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
22.
Solution:
It is given that
∫ sin x log (cos x) dx
Consider first function as log (cos x) and second function as sin x
So we get
= – cos x log (cos x) – ∫ sin x dx
Again by integrating the second term
= – cos x log (cos x) + cos x + c
23.
Solution:
It is given that
∫ x sin x cos x dx
We know that
sin 2x = 2 sin x cos x
It can be written as
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
Consider first function as x and second function as sin 2x
24.
Solution:
It is given that
∫ cos √x dx
Take √x = t
So we get
1/2√x dx = dt
By cross multiplication
dx = 2 √x dt
Here dx = 2t dt
It can be written as
∫ cos √x dx = 2 ∫t cos t dt
Take first function as t and second function as cos t
By integrating w.r.t t
= 2 (t sin t – ∫ sin t dt)
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
We get
= 2t sin t + 2 cos t + c
Now by substituting t as √x
= 2√x sin √x + 2 cos √x + c
Taking 2 as common
= 2 (cos √x + √x sin √x) + c
25.
Solution:
It is given that
∫ cosec3 x dx
We can write it as
∫ cosec3 x dx = ∫ cosec x cosec2 x dx
So we get
By integration we get
= cosec x (- cot x) – ∫ (- cosec x cot x) (- cot x) dx
It can be written as
= – cosec x cot x – ∫ cosec x cot2 x dx
Here cot2 x = cosec2 x – 1
We get
= – cosec x cot x – ∫ cosec x (cosec2 x – 1) dx
By further simplification
= – cosec x cot x – ∫ cosec3 x dx + ∫ cosec x dx
We know that ∫ cosec3 x dx = 1
∫cosec3 x dx = – cosec x cot x – ∫ cosec3 x dx + ∫ cosec x dx
On further calculation
2 ∫cosec3 x dx = – cosec x cot x + ∫ cosec x dx
By integration we get
2 ∫cosec3 x dx = – cosec x cot x + log |tan x/2| + c
Here
∫cosec3 x dx = – ½ cosec x cot x + ½ log |tan x/2| + c
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
27.
Solution:
It is given that
∫ sin x log (cos x) dx
Take cos x = t
So we get
– sin x dx = dt
It can be written as
∫ sin x log (cos x) dx = – ∫log t dt = – ∫1. log t dt
Consider first function as log t and second function as 1
By integrating w.r.t t
= – log t. t + ∫1/t. t dt
Again by integrating the second term
= – t log t + t + c
Now replace t as cos x
= t (- log t + 1) + c
We get
= cos x (1 – log (cos x)) + c
28.
Solution:
Take log x = t
So we get
1/x dx = dt
Here
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
Again by integrating the second term we get
= t log t – t + c
Replace t with log x
= log x. log (log x) – log x + c
By taking log x as common
= log x (log (log x) – 1) + c
29.
Solution:
It is given that ∫ log (2 + x2) dx
We can write it as
= ∫1. log (2 + x2) dx
Consider first function as log (2 + x2) and second function as 1
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
So we get
31.
Solution:
Consider log x = t
Where x = et
So we get
dx = et dt
We can write it as
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
32.
Solution:
Using the formula
cos A cos B = ½ [cos (A + B) + cos (A – B)]
We get
cos 4x cos 2x = ½ [cos (4x + 2x) + cos (4x – 2x)] = ½ [cos 6x + cos 2x]
By applying in the original equation
∫ e –x cos 2x cos 4x dx = ∫ e –x (½ [cos 6x + cos 2x])
= ½ [∫e –x cos 6x dx + ∫e –x cos 2x dx]
Taking first function as cos 6x and cos 2x and second function as e –x
Now by solving both parts separately
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
33.
Solution:
Consider √x = t
We get
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
It can be written as
dx = 2 √x dt
where dx = 2t dt
We can replace it in the equation
∫e √x dx = ∫et 2t dt
So we get
= 2 ∫t et dt
Consider the first function as t and second function as et
By integration w.r.t. t
= 2 (t et – ∫1. et dt)
We get
= 2 (t et – et) + c
Taking et as common
= 2 et (t – 1) + c
Substituting the value of t we get
= 2 e √x (√x – 1) + c
34.
Solution:
We know that
sin 2x = 2 sin x cos x
So we get
∫ e sin x sin 2x dx = 2 ∫ e sin x sin x cos x dx
Take sin x = t
So we get cos x dx = dt
It can be written as
2 ∫ e sin x sin x cos x dx = 2 ∫ e t t dt
Consider first function as t and second function as et
RS Aggarwal Solutions for Class 12 Maths Chapter 13
Methods of Integration
By integrating w.r.t. t
= 2 (t et – ∫1. et dt)
We get
= 2 (t et – et) + c
Here
= 2 et (t – 1) + c
By substituting the value of t
= 2 esin x (sin x – 1) + C
35.
Solution:
Take sin -1 x = t
Here x = sin t
So we get
By integration we get
= t (- cos t) – ∫1. (- cos t) dt
We get
= – t cos t + sin t + c
It can be written as
cos t = √1 – sin2 t
Here we get
= – t (√1 – sin2 t) + sin t + c
By replacing t as sin-1 x
= – sin-1 x (√1 – x2) + x + c