Class 12 RD Sharma Solutions – Chapter 20 Definite Integrals – Exercise 20.1 | Set 1
Evaluate the following definite integrals:
Question 1.
Solution:
We have,
I =
I =
I =
I =
I = 2[√9 – √4 ]
I = 2 (3 − 2)
I = 2 (1)
I = 2
Therefore, the value of is 2.
Question 2.
Solution:
We have,
I =
I =
I = log (3 + 7) − log (−2 + 7)
I = log 10 − log 5
I =
I = log 2
Therefore, the value of is log 2.
Question 3.
Solution:
We have,
I =
Let x = sin t, so we have,
=> dx = cos t dt
Now, the lower limit is,
=> x = 0
=> sin t = 0
=> t = 0
Also, the upper limit is,
=> x = 1/2
=> sin t = 1/2
=> t = π/6
So, the equation becomes,
I =
I =
I =
I =
I =
I = Ï€/6 – 0
I = π/6​
Therefore, the value of is π/6.
Question 4.
Solution:
We have,
I =
I =
I =
I =
I = π/4
Therefore, the value of is π/4.
Question 5.
Solution:
We have,
I =
Let x2 + 1 = t, so we have,
=> 2x dx = dt
=> x dx = dt/2
Now, the lower limit is, x = 2
=> t = x2 + 1
=> t = (2)2 + 1
=> t = 4 + 1
=> t = 5
Also, the upper limit is, x = 3
=> t = x2 + 1
=> t = (3)2 + 1
=> t = 9 + 1
=> t = 10
So, the equation becomes,
I =
I =
I =
I = 1/2[log10 – log5]
I = 1/2[log10/5]
I = 1/2[log2]
I = log√2
Therefore, the value of is log√2.
Question 6.
Solution:
We have,
I =
I =
I =
I =
I =
I = 1/ab[tan-1∞ – tan-10]
I = 1/ab[Ï€/2 – 0]
I = 1/ab[Ï€/2]
I = π/2ab
Therefore, the value of is π/2ab.
Question 7.
Solution:
We have,
I =
I =
I = [tan-11 – tan-1(-1)]
I = [Ï€/4 – (-Ï€/4)]
I = [π/4 + π/4]
I = 2Ï€/4
I = π/2
Therefore, the value of is π/2.
Question 8.
Solution:
We have,
I =
I =
I = -e–∞ – (-e0)
I = − 0 + 1
I = 1
Therefore, the value of is 1.
Question 9.
Solution:
We have,
I =
I =
I =
I =
I =
I = [1 − 0] − [log(1 + 1) − log(0 + 1)]
I = 1 − [log2 − log1]
I = 1 – log2/1
I = 1 − log 2
I = log e − log 2
I = loge/2
Therefore, the value of is loge/2.
Question 10.
Solution:
We have,
I =
I =
I =
I = [-cosÏ€/2 + cos0] + [sinÏ€/2 – sin0]
I = [−0 + 1] + 1
I = 1 + 1
I = 2
Therefore, the value of is 2.
Question 11.
Solution:
We have,
I =
I =
I = log(sinÏ€/2) – log(sinÏ€/4)
I = log1 – log1/√2
I =
I = log√2
Therefore, the value of is log√2.
Question 12.
Solution:
We have,
I =
I =
I = log(secÏ€/4 + tanÏ€/4 – log(sec0 + tan0)
I = log(√2 + 1) – log(1 + 0)
I =
I = log(√2 + 1)
Therefore, the value of is log(√2 + 1).
Question 13.
Solution:
We have,
I =
I =
I = [log|cosecÏ€/4 – cotÏ€/4|] – [log|cosecÏ€/6 – cotÏ€/6|]
I = [log|√2 – 1|] – [log|2 – √3|]
I =
Therefore, the value of is .
Question 14.
Solution:
We have,
I =
Let x = cos 2t, so we have,
=> dx = –2 sin 2t dt
Now, the lower limit is,
=> x = 0
=> cos 2t = 0
=> 2t = π/2
=> t = π/4
Also, the upper limit is,
=> x = 1
=> cos 2t = 1
=> 2t = 0
=> t = 0
So, the equation becomes,
I =
I =
I =
I =
I =
Let cos t = z, so we have,
=> – sin t dt = dz
=> sin t dt = – dz
Now, the lower limit is,
=> t = 0
=> z = cos t
=> z = cos 0
=> z = 1
Also, the upper limit is,
=> t = π/4
=> z = cos t
=> z = cos π/4
=> z = 1/√2
So, the equation becomes,
I =
I =
I =
I =
I =
I =
I = -4[(log1/√2 – 1/2(2)) – (log1 – 1/2)]
I = -4[(log1/√2 – 1/4) – (0 – 1/2)]
I = -4[log1/√2 – 1/4 – 0 + 1/2]
I = -4[-log√2 + 1/4]
I = 4log√2 – 1
I = 4 × 1/2log2 – 1
I = 2log2 – 1
Therefore, the value of is 2log2 – 1.
Question 15.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I =
I =
I =
I = [tan π – tan0] – [sec π – sec 0]
I = [0 – 0] – [–1 – 1]
I = 0 – (–2)
I = 2
Therefore, the value of is 2.
Question 16.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I =
I =
I =
I =
I = [tan π/4 – tan(–π/4)] – [sec π/4 – sec (–π/4)]
I = [1 – (–1)] – [sec π/4 – sec (π/4)]
I = 2 – 0
I = 2
Therefore, the value of is 2.
Question 17.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I = 1/2[Ï€/2 – 0] + 1/4[sinÏ€ – sin0]
I = 1/2[Ï€/2] + 1/4[0 – 0]
I = π/4
Therefore, the value of is π/4.
Question 18.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I = 1/12 [-1 – 0] + 3/4[1 – 0]
I = 3/4 – 1/12
I = (9 – 1)/12
I = 8/12
I = 2/3
Therefore, the value of is 2/3.
Question 19.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I = 1/6[sinÏ€/2 – sin0] + 1/2[sinÏ€/6 – sin0]
I = 1/6[1 – 0] + 1/2[1/2 – 0]
I = 1/6 + 1/4
I = (4 + 6)/24
I = 10/24
I = 5/12
Therefore, the value of is 5/12.
Question 20.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I = 1/2[sinÏ€/2 – sin0] – 1/6[sin3Ï€/2 – sin0]
I = 1/2[1 – 0] – 1/6[-1 – 0]
I = 1/2 – 1/6(-1)
I = 1/2 + 1/6
I = (6 + 2)/12
I = 8/12
I = 2/3
Therefore, the value of is 2/3.
Question 21.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I =
I =
I =
I =
I = 2[-cotπ/2 + cot2π/3]
I = 2[-1/√3 – 0]
I = -2/√3
Therefore, the value of is -2/√3.
Question 22.
Solution:
We have,
I =
I =
I =
I =
I =
I =
I =
I = 1/4[Ï€/2 + Ï€/4 + 0 + 0 – 0 – 0 – 0 – 0]
I = 1/4[3Ï€/4]
I = 3Ï€/16
Therefore, the value of is 3Ï€/16.
Last Updated :
02 Jun, 2021
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