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Textbook | NCERT |

Class | Class 12th |

Subject | Maths |

Chapter | 20 |

Exercise | 20.3 |

Category | RD Sharma Solutions |

Table of Contents

**RD Sharma Class 12 Ex 20.3 Solutions Chapter 20 Definite Integrals**

**Question 1: Ten cards numbered 1 through 10 are placed in a box, mixed **

### Evaluate the following integrals:

### Question 1(i).

**Solution:**

We have,

I =

I =

Using additive property, we get

I =

I =

I = 8 + 6 – 2 – 3 + 24 + 20 – 6 – 10

I = 37

### Question 1(ii).

**Solution:**

We have,

I =

I =

Using additive property, we get

I =

I =

I =

I = 0 + 1 + 3 – π/2 + e

^{6}– e^{0}I = 3 – π/2 + e

^{6}

### Question 1(iii).

**Solution:**

We have,

I =

I =

Using additive property, we get

I =

I =

I =

I = 63/2 + 9 – 7/2 – 3 + 64 – 36

I = 56/2 + 34

I = 62

### Question 2.

**Solution:**

We have,

I =

We know that,

So, we get

I =

I =

I =

I = –2 + 4 – 8 – 8 + 8 + 8 – 2 + 4

I = 20

### Question 3.

**Solution:**

We have,

I =

We know that,

So, we get

I =

I =

I = 0 + 2 + 8 – 0

I = 10

### Question 4.

**Solution:**

We have,

I =

We know that,

So, we get

I =

I =

I = –1/4 + 1/2 + 1 – 1 + 1 + 1 – 1/4 + 1/2

I = 5/2

### Question 5.

**Solution:**

We have,

I =

We know that,

So, we get

I =

I =

I = –9/4 + 9/2 + 4 – 6 + 4 + 6 – 9/4 + 9/2

I = 25/2

### Question 6.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = 1/3 – 3/2 + 2 – [8/3 – 6 + 4 – 1/3 + 3/2 – 2]

I = 1/3 – 3/2 + 2 – 8/3 + 6 – 2 + 1/3 – 3/2

I = 1

### Question 7.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I = –1/6 + 1/3 – 0 + 27/2 + 3 – 1/6 – 1/3

I = 65/6

### Question 8.

**Solution:**

We have,

I =

We know that,

So, we get

I =

I =

I =

I = –2 + 4 + 18 – 12 + 18 + 12 – 2 + 4

I = 40

### Question 9.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = -1/2 + 1 + 2 – 2 + 2 + 2 – 1/2 + 1

I = 5

### Question 10.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = – 2 – 6 + 1/2 + 3

I = – 5 + 1/2

I = (-10 + 1)/2

I = -9/2

### Question 11.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = 1/2 – 0 – 0 + 1/2

I = 1

### Question 12.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = 1 + 1 + 1 – (–1)

I = 1 + 1 + 1 + 1

I = 4

### Question 13.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = 1 – 1/√2 – 1/√2 + 1

I = 2 – 2/√2

I = 2 – √2

### Question 14.

**Solution:**

We have,

I =

We know that,

So we get,

I =

I =

I =

I = – 25/2 + 25 + 2 – 10 + 32 – 40 – 25/2 + 25

I = 9

### Evaluate the following integrals:

### Question 15.

**Solution:**

We have,

I =

Since f(- x) = sin|- x| + cos|- x|

= sin |x| + cos |x|

= f(x)

So, f(x) is an even function.

Therefore, we get

I =

I =

I =

I = 2 (0 + 1 + 1 – 0)

I = 4

### Question 16.

**Solution:**

We have,

I =

We know,

So we get,

I =

I =

I =

I = -1/2 + 1 – 0 + 8 – 4 – 1/2 + 1

I = 5

### Question 17.

**Solution:**

We have,

I =

We know,

So we get,

I =

I =

I = 8 – 4 – 1/2 + 1 – (2 – 4 – 1/2 + 2) + 8 – 8 – 2 + 4 – (8 – 16 – 1/2 + 4)

I = 23/2

### Question 18.

**Solution:**

We have,

I =

We know,

So we get,

I =

I =

I = 25/2 – (2 – 4 – 25/2 + 10) – 2 + 4 + (-25/2 + 25)

I = 63/2

### Question 19.

**Solution:**

We have,

I =

We know,

So we get,

I =

I =

I = 8 – (2 – 4) + 8 – 8 – 2 + 4 – (8 – 16)

I = 20

### Question 20.

**Solution:**

We have,

I =

We know,

When –1 < x < 0,

|x + 1| + |x| + |x – 1| = x + 1 + (- x) + [-(x – 1)]

= 2 – x

And when 0 < x < 1,

|x + 1| + |x| + |x – 1| = x + 1 + x + [-(x – 1)]

= x + 2

And when 1 ≤ x ≤ 2,

|x + 1| + |x| + |x – 1| = x + 1 + x + x – 1

= 3x

So we get,

I =

I =

I =

I = – 1/2(4 – 9) + 1/2( 9 – 4) + 3/2(4 – 1)

I = 5/2 + 5/2 + 9/2

I = 19/2

### Question 21.

**Solution:**

We have,

I =

Now here,

f(- x) = (- x)e

^{|- x|}= – x e

^{|x|}= – f(x)

So, f(x) is an odd function.

Therefore we get,

I =

I = 0

### Question 22.

**Solution:**

We have,

I =

I =

As we know,

I =

I =

I =

I =

I =

I = -1/2(0 + π/4) + 1/4(0 + sin π/2) + 1/2 ( π/2 – 0) – 1/4(sin π – 0)

I = – π/8 + 1/4 (0 + 1) + π/8 – 1/4 (0 – 0)

I = π/8 + 1/4

### Question 23.

**Solution:**

We have,

I =

Now here,

f(π – x) = cos(π – x)|cos(π – x)|

= -cos x|-cos x|

= – cos x|cos x|

= – f(x)

So, f(x) is an odd function.

Therefore we get,

I =

I = 0

### Question 24.

**Solution:**

We have,

I =

Now here,

f(- x) = 2sin|- x| + cos|- x|

= 2sin|x| + cos|x|

= f(x)

So, f(x) is an odd function.

Therefore we get,

I =

I =

As we know,

I =

I =

I =

I = – 4(cos π/2 – cos 0) + 2(sin π/2 – sin 0)

I = –4 ( 0 – 1) + 2 (1 – 0)

I = 4 + 2

I = 6

### Question 25.

**Solution:**

We have,

I =

I =

I =

As π/2 ≤ x ≤ π, we get

=> –π ≤ –x ≤ –π/2

=> 0 ≤ π – x ≤ π/2

So, we get

I =

I = 1/2 (π

^{2}/4 – π^{2}/4) – 1/2( 0 – π^{2}/4)I = 0 + π

^{2}/8I = π

^{2}/8

### Question 26.

**Solution:**

We have,

I =

I =

I =

As we know,

= √cos x|-sin x|

= √cos x|sin x|

= f(x)

So, f(x) is an odd function.

Therefore we get,

I =

I =

As we know, ,

I =

I =

Let cos x = z

^{2}. So, we have=> – sin x dx = 2z dz

Now, the lower limit is, x = 0

=> z

^{2}= cos x=> z

^{2}= cos 0=> z

^{2}= 1=> z = 1

Also, the upper limit is, x = π/2

=> z

^{2}= cos x=> z

^{2}= cos π/2=> z

^{2}= 0=> z = 0

So, the equation becomes,

I =

I =

I =

I =

I =

I = – π/2(log1 – log0) + π/2(log1 – log2) + π(tan

^{-1}0 – tan^{-1}1)I = – π/2[0 – ∞] + π/2(0 – log2) + π(0 – π/4)

I = -∞ – π/2 log2 – π

^{2}/4I = –∞

### Question 27.

**Solution:**

We have,

I =

I =

As we know,

I =

I =

I =

I = 4 – 1

I = 3

### Question 28.

**Solution:**

We have,

I =

I =

As we know, π ≤ x ≤ 2π

=> –2π ≤ –x ≤ –π

=> 0 ≤ 2π – x ≤ π

Therefore, we get

I =

I =

I = 1/2( π – 0) – 1/2(0 – π)

I = π

^{2}/2 + π^{2}/2I = π

^{2}

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