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Kathmandu University
Assignment-III
Subject: Calculus and Linear Algebra (MATH 101 and MATH111 )
Issued Date: 2/12/2018 Deadline of submission: 20/12/2018(Strictly
Follow the date line)
Task 1: (Indefinite integrals)
- Define indefinite integral of a function f(x). Evaluate the following:
(i) sin xdx
4
(ii) (4secx tanx - 2sec )dx
2
x (iii) (e ) dx
ax 3
bx e
(iv) dx 2
x 5
2
x
(v) (2x 3)(4x 7) dx
3 (^) (vi) dx (x 2)
2x 3 4
(vii) (^) ^1 - cosxdx (viii) 3x^1 x dx
2 3 (ix) dt t 2t 2
t 1 (^3 )
(x) dx x
cot x
(xi) secxtanxdx
3
(xii) x(1 x)
dx
(xiii) sin(lnx)dx x
(^) (xiv) tan xsecxdx
3 4 (^) (xv) dx 1
sin(tan ) 2
x
x
- Evaluate the following:
(i) dx x 9
(^2)
(ii) dx x 4
2
(iii) dx (x 1)
x 2
(iv)
2 2 x 1 x
dx (v) dx 9 x
x
2
2
(vi) xsin2xdx
3
(vii) x lnxdx
2 (^) (viii) cosec dx
3 (^) x (ix) a x dx
2 2
(x) e dx
x
(xi) e cosbxdx
ax
(xii) sin xdx
1
(xiii) (^) cos(logx)dx (xiv) xsin xdx
1
(xv) cos x dx
- Evaluate the following:
(i) x 49
dx 2
(ii) 2 4 - 5x
dx (iii) x 2x 5
dx 2
(iv)
x 4x 5
(2x 3)dx 2
(v)
9x 6x 26
(6x 2)dx 2 (vi) 36x 121
dx
2
(vii)
x 1
(x 2)dx
2 x
(viii)
2 2x x
dx (ix) dx 8 2x x
1 x (^2)
(x) x dx
2 25 9
(xi) ax x dx
2 2 (xii) ( x 2 ) x 10 x 11 dx
2 (xiii) x x x dx
2 ( 2 ) 16 6
(xiv) x 4
xdx
2
3 (xv) ( 1 )
x dx 2
2
x
- Evaluate following:
(i)
dx (x 3)(x 2)
5x 13 (ii)
dx (x 1)
x 4 2
iii)
dx x (x 1)
x 1 2
iv)
dx x 5x 6
x 8 2
2
(v) (x 1)(x 1)
dx 2 (vi) 1 sinx
dx (vii) sinx cosx
dx (viii) 2 cosx
dx
(ix) a sin x cosx
dx 2 2 2 2 b
(x) 4 - 5sin x
dx 2
Task 2: (Definite Integrals)
- Evaluate the following definite integrals:
(i)
(^3 )
(t 1)(t 4) dt
(ii)
2
--1 2
tdt
2t +
(iii)
π (^3 )
-2π 3
x x tan sec dx 4 4
(iv)
-1 2 (^) -2 2
--
t sin (1+ ) dt t
(v)
(^1 2 3) -2 3 2
(4y-y +4y +1)^ (12y -2y+4) dy
(vi)
2
2
π 4
π 36
cos t dt
tsin t
(vii)
4
2 2
dx
x(lnx)
(viii)
π
π 2
θ 2cot dθ 3
(ix)
π 4 (^) tanθ 2
0
(1+e )sec θ dθ
(x)
lnπ (^) x 2 x 2
0
2xe cos(e ) dx
(xi)
--1 2
dt
t 4t -
(xii)
2
1 2
8dx
x -2x+
(xiii)
e (^3)
1
x lnx dx (xiv)
(^1 2) -1 2
0
2xsin (x ) dx
(xv)
1
0 2
dx
(x+1)(x +1)
(xvi)
3 2
1 3
(3x +x+4) dx
(x +x)
(xvii)
π 2
0
dθ
2+cosθ
- State the Fundamental Theorems of calculus part first and second.
- Find , if dx
dy (i)y e dt
x^2
2
t
(ii) dt 1 t
tan t y
0
tanx
2
1
(iii)y sint dt
2 x
x
2
- Evaluate following definite integrals
(i)
2
1
x x 2 dx
2 (ii)
π
0 1 sinx
dx (iii)
, if x 2 2
x
3 x, ifx 2
f(x)dx wheref(x)
3
1
(iv)
2
2
2 x x dx
Task 3: (Improper Integrals) I. Define improper integrals. Investigate the convergence of the following improper integrals :
(i)
0
2
(ii)
2
4 - x
dx
(iii)
xe dx
(iv)
^
2 1 x
dx
(v)
1
0 x
dx
(vi)
6
4 x-^4
dx
(vii)
2
0
(^3) (x 1)
dx (viii)
1
1
3 x
dx (ix)
1
1
3 x
dx
Task 4 :(Average Value/Initial Value Theorem)
- Find the average values of the following functions on the given intervals and verify the result:
(i) f(x)= 2
π sinx, 0 x (ii) f(x)= 2x 1 , 4 x 12 (iii)
- Solve the following initial value problems:
(i) ;y(0)^0 1 x
dx
dy
2
(ii) ;y(0) 0 x 1
x
dx
dy 2
(iii) 2xy ;y(1) 1 dx
dy (^2)
(iv) (v)
Task 5:(Area as a limit of a sum)
- Find the area of the region between the curve and the x-axis on the interval [2,5] using limit of a sum using i) Inscribed rectangles ii) Circumscribed rectangles
- Calculate the area from first principle(definition) or Using method of summation evaluate the following:
i)
3b (^3)
1
x dx ii)
2
1
x dx
iii)
1 2 (^2)
0
(x -4x+5)dx
iv)
5
1
x ( +1) dx 2
v)
(^5 3 )
1
(x -3x +5x-20) dx
