第 12 类 RD Sharma 解 – 第 20 章定积分 – 练习 20.1 |设置 3
计算以下定积分:
问题 45。
解决方案:
We have,
I =
Let 2x + 1 = t2, so we have,
=> 2 dx = 2t dt
=> dx = t dt
Now, the lower limit is, x = 1
=> t2 = 2x + 1
=> t2 = 2(1) + 1
=> t2 = 3
=> t = √3
Also, the upper limit is, x = 4
=> t2 = 2x + 1
=> t2 = 2(4) + 1
=> t2 = 9
=> t = 3
So, the equation becomes,
I =
I =
I =
I =
I =
I = 1/4 [35/5 – 3 – (√3)5/5 + √3]
I = 1/4[243/5 – 3 – 9√3/5 + √3]
I = 1/4((243 – 15 – 9√3 + 5√3)/5)
I = 1/4[(228 – 4√3)/5]
I = 1/4[4(57 – √3)/5]
I = (57 – √3)/5
Therefore, the value of is (57 – √3)/5.
问题 46。
解决方案:
We have,
I =
By using binomial theorem in the expansion of (1 – x)5, we get,
I =
I =
I =
I =
I = 1/2 – 5/3 + 10/4 – 10/5 + 5/6 – 1/7
I = 1/2 – 5/3 + 5/3 – 2 + 5/6 – 1/7
I = 1/2 – 2 + 5/6 – 1/7
I = 1/42
Therefore, the value of is 1/42.
问题 47。
解决方案:
We have,
I =
I =
I =
I =
By using integration by parts, we get,
I =
I =
I = ex/x
So we get,
I =
I = e2/2 – e1/1
I = e2/2 – e
Therefore, the value of is e2/2 – e.
问题 48。
解决方案:
We have,
I =
By using integration by parts in first integral, we get,
I =
I = xe2x/2 – (1/2)(e2x/2) + 2/π[1 – 0]
I = xe2x/2 – e2x/4 + 2/π
So we get,
I =
I = [e2/2 + e2/4 – 0 + 1/4] + 2/π
I = e2/4 + 1/4 + 2/π
Therefore, the value of is e2/4 + 1/4 + 2/π.
问题 49。
解决方案:
We have,
I =
By using integration by parts in first integral, we get,
I =
I =
I =
I =
So we get,
I =
I =
I = [e1(1 – 1) – e0(0 – 1)] + 2√2/π
I = [0 – (-1)] + 2√2/π
I = 1 + 2√2/π
Therefore, the value of is 1 + 2√2/π.
问题 50。
解决方案:
We have,
I =
I =
I =
I =
I = -eπ cotπ/2 + eπ/2 cotπ/4
I = 0 + eπ/2(1)
I = eπ/2
Therefore, the value of is eπ/2.
问题 51。
解决方案:
We have,
I =
I =
I =
I =
By using integration by parts in first integral, we get,
I =
I =
I =
I = 1/√2[sinπ(2eπ) – 0]
I = 1/√2[0 – 0]
I = 0
Therefore, the value of is 0.
问题 52。
解决方案:
We have,
I =
By using integration by parts, we get,
I = excos(x/2 + π/4) + 1/2∫exsin(x/2 + π/4)
I = ex cos(x/2 + π/4) + 1/2[ exsin(x/2 + π/4) – 1/2 ∫excos(x/2 + π/4)dx]
I = excos(x/2 + π/4) + 1/2exsin(x/2 + π/4) – 1/4I
5I/4 = -3/ 2√2(e2π + 1)
I = -3√2/5(e2π + 1)
Therefore, the value of is -3√2/5(e2π + 1).
问题 53。
解决方案:
We have,
I =
I =
I =
I =
I =
I =
I = 2/3[23/2 – 1] + 2/3[1 – 0]
I =
I = 25/2/3
Therefore, the value of is 25/2/3.
问题 54。
解决方案:
We have,
I =
I =
I =
I =
I = -log3 + log2 + 2[log4 – log3]
I = -log3 + log2 + 2[2log2 – log3]
I = -log3 + log2 + 4log2 – 2log3
I = 5log2 – 3log3
I = log25 – log33
I = log32 – log27
I = log32/27
Therefore, the value of is log32/27.
问题 55。
解决方案:
We have,
I =
I =
I =
Let cos x = t, so we have,
=> – sin x dx = dt
Now, the lower limit is, x = 0
=> t = cos x
=> t = cos 0
=> t = 1
Also, the upper limit is, x = π/2
=> t = cos x
=> t = cos π/2
=> t = 0
So, the equation becomes,
I =
I =
I =
I = [0 – 1/3] – [0 – 1]
I = [-1/3] – [-1]
I = -1/3 + 1
I = 2/3
Therefore, the value of is 2/3.
问题 56。
解决方案:
We have,
I =
I =
I =
I =
I = -sinπ + sin0
I = 0
Therefore, the value of is 0.
问题 57。
解决方案:
We have,
I =
Let 2x = t, so we have,
=> 2x dx = dt
Now, the lower limit is, x = 1
=> t = 2x
=> t = 2(1)
=> t = 2
Also, the upper limit is, x = 2
=> t = 2x
=> t = 2(2)
=> t = 4
So, the equation becomes,
I =
I =
I =
By using integration by parts in first integral, we get,
I =
I =
I =
I = e4/4 – e2/2
Therefore, the value of is e4/4 – e2/2.
问题 58。
解决方案:
We have,
I =
I =
I =
I =
I =
I =
I = [sin-1(1) – sin-1(-1)]
I = π/2 – (-π/2)
I = π/2 + π/2
I = π
Therefore, the value of is π.
问题 59. 如果 ,求k的值。
解决方案:
We have,
=>
=>
=>
=>
=> tan-12k/4 – tan-10 = π/16
=> tan-12k/4 – 0 = π/16
=> tan-12k/4 = π/16
=> tan-12k = π/4
=> 2k = tanπ/4
=> 2k = 1
=> k = 1/2
Therefore, the value of k is 1/2.
问题 60. 如果 ,求k的值。
解决方案:
We have,
=>
=>
=>
=>
=> a3 – 0 = 8
=> a3 = 8
=> a = 2
Therefore, the value of a is 2.
问题 61。
解决方案:
We have,
I =
I =
I =
I =
I = -[√2cos3π/2 – √2cosπ]
I = -(-√2 – 0)
I = √2
Therefore, the value of is √2.
问题 62。
解决方案:
We have,
I =
I =
I =
I =
I =
I =
I =
I = [-4cosπ/2 + 4cos0] + [4sinπ/2 – 4sin0]
I = 0 + 4 + 4 – 0
I = 8
Therefore, the value of is 8.
问题 63。
解决方案:
We have,
I =
I =
I =
I =
I =
I =
I =
I =
I =
I =
I =
I = (π/8 – 1/4) – (3/4(π/8 – 1/4) – 1/16)
I = π/8 – 1/4 – (3π/32 – 3/16 – 1/16)
I = π/8 – 1/4 – (3π/32 – 1/4)
I = π/8 – 1/4 – 3π/32 + 1/4
I = π/8 – 3π/32
I = (4π – 3π)/32
I = π/32
Therefore, the value of is π/32.
问题 64。
解决方案:
We have,
I =
By using integration by parts we get,
I =
I =
I =
I =
I =
So we get,
I =
I = log3/2 – 1/8log3
I = 3/8log3
Therefore, the value of is 3/8log3.
问题 65。
解决方案:
We have,
I =
I =
I =
I =
I =
I =
I =
I = [tanπ/3 – tanπ/6] + [-cotπ/3 + cotπ/6]
I = [√3 – 1/√3] + [- 1/√3 – √3]
I = 2[√3 – 1/√3]
I = 4/√3
Therefore, the value of is 4/√3.
问题 66。
解决方案:
We have,
I =
I =
I =
I =
I =
I =
I =
I =
I =
I =
I =
Therefore, the value of is .
问题 67。
解决方案:
We have,
I =
I =
I =
I =
I = -log2/4 + log2/2 – 1/4 + 1/2
I = log2/4 + 1/4
Therefore, the value of is log2/4 + 1/4.