INTEGRAL CALCULAS

Answer to the Question No: 1

Ques: The maintenance per month for a factory which is t months old is 2000+100t. What is the maintenance cost of the factory for its (a) first year (b) tenth year.
Solution: (¡) The cost for the first year
012ctdt=0122000t+10tdt
=2000t+10t22120
=2000t+5t²120
=2000×12+5×12²
=TK.24720
(¡¡) The cost for the tenth year
108120ctdt=1081202000t+10tdt
=2000t+5t²120108
=2000×120+5120²-2000×108+5108²
= 312000-274320
=TK.37680
Answer: TK.24720, TK.37680
Answer to the Question No: 2
Ques: The marginal cost function for a certain commodity is MC=3q²─4q+5.Determine the cost of producing the eleventh through the fifteenth units, inclusive.
Solution: The cost of 15 units=Q³─2Q²+5Q
= (15)³─2 (15)²+5(15)
=3000
The cost of 10 units= (10)³─2 (10)²+5(10)
=850
The difference between the cost of 15 units & 10 units indicates the incremental cost of producing the 11th to the 15th units of output.
Thus, TC (15) ─TC (10) =3000─850
=2150
Answer: 2150
Answer to the Question No: 3
Ques: Integrate the 2x²-5xdx
Solution:
2x²-5xdx
=2x-5xdx
=2xdx-5 1xdx
=2xdx-51xdx
=2x²2-5logx
=x²-5 logx
Ans:
Answer to the Question No: 4
Ques: Evaluate the following definite integrals:
(¡) 0adxa²+x² (¡¡) 01tan-1x²1+x²dx
Solution: (¡) 0adxa²+x²
= 1atan-1xaa0
= 1atan-1aa-1atan-10a
= 1atan-11-1atan-10
=1a π4-1a∙0
=π4a
Ans
(¡¡) 01tan-1x²1+x²dx Let,
= 01t²dt tan-1x=t
=t³310 ⇒ddxtan-1x=ddxt
=1³3 ⇒11+x²=dtdx
=13 ⇒11+x²dx=dt

Ans:
Answer to the Question: 5
Ques: The revenue on the t th business day at a grocery store is 5000+3t. How much is the total revenue for the first hundred business days? for the second hundred business days?
Solution: Let R (t) is the total revenue for the first t days of business. Then R` (t) is the revenue on the t th day of business, so that R` (t)= 500+3t. The total revenue for the first hundred business days is
0100R´tdt=0100500+3tdt
=500t+3t²21000
=500×100+3100²2
=TK. 65000.
The revenue for the second hundred business days is
100200R´tdt=100200500+3tdt
=500t+3t²2200100
= (1, 00,000+60,000)-(50,000+15,000)
=TK. 95,000.
Ans:
Answer to the Question No:6
Ques: 2x-1xx-1x-2
Solution: 2x-1xx-1x-2
Let, 2x-1xx-1x-2≡Ax+Bx-1+Cx-2
⇒2x-1=Ax-1x-2+Bxx-2+
Cxx-2...................(1)
When, x=1,then(1)⇒2.1-1=A(1-1)(1-2)+B11-2+C11-1
⇒1= -B
⇒B=1
When, x=2,then(1)⇒ 2.2-1=A1-22-2+B22-2+C22-1
⇒3=2c
⇒c= 32
So,0 =A+B ∴A=-B=1
∴2x-1xx-1x-2dx
=[1x+-1x-1+32x-2]dx
=logx-log(x-1)+ 3 2log(x-2)+c
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