Detailed Solutions for Calculus and Analysis Homework Assignment

This document presents solutions to a series of Calculus and Analysis homework problems. It includes finding the Maclaurin series expansion for ln(x+1), solving limit problems using L'Hopital's rule, evaluating indefinite and definite integrals, and addressing a probability question related to stochastic processes. The solutions demonstrate step-by-step calculations and explanations for each problem, covering topics such as derivatives, integration techniques, and the application of mathematical laws in probability.

Ans 4. L n ( x ) =∑
n=1
∞ (−1 )n−1 ( x−1 )n
n
¿
¿
ln ( x +1 ) =∑
n=1
∞ ( −1 ) n−1 ( x ) n
n
¿
¿
xn−1 ln ( x +1 ) =xn−1
∑
n=1
∞ ( −1 ) n −1 ( x ) n
n =¿ xn−1
∑
k=1
n ( −1 ) n−1 ( x ) n
( 1+x ) k =¿ ( n−1 ) !∑
k=1
n 1
( 1+ x ) k
¿
¿
Ans 5.
1. By the definition of derivative,
Lim (f(x) - f(x0))/x-x0 =f’(x)
x-> x0
Lim (g(x) - g(x0))/g-g0 =g’(x)
x-> x0
lim f(x)/g(x) = lim ((f(x) - f(x0))/x-x0)/( (g(x) - g(x0))/g-g0)= f’(x)/ g’(x) > 1
x-> x0 x->x0
2.
lim
n→ 0 ( 1−cos ( x )
x2 )=
2 sin2
( x
2 )
x2
=
2
4 ( sin2
( x
2 )
x
2
2
)=
1
2∗1 = 1/ 2
Ans 6.
1. ∫ x3 ln x dx
∫ x3 lnxdx=lnx( x4
4 )−∫ ( x4
4 )( 1
x ) dx
¿ x4 lnx
4 − 1
4 ∫ x3 dx
¿ x4 lnx
4 − x4
16 +C

¿ x4 ( 4 lnx−1 )
16 +C
2. ∫ xarctan ( x ) dx
¿ ∫ d
dx ( x2
2 ) arctanx dx
¿ ( x2
2 )arctanx− ∫ x2
2
d
dx (arctanx )dx
¿ ( x2
2 )arctanx− ∫ x2
2 ⋅ 1
x2 +1 dx
¿ ( x2
2 )arctanx− 1
2 ∫ x2
x2 +1 dx
¿ ( x2
2 ) arctanx− 1
2 ∫ (x¿ ¿ 2+1−1)/(x¿ ¿ 2+1)dx ¿ ¿
¿( x2
2 ) arctanx− 1
2 ∫ 1− 1
x2+ 1 dx
¿( x2
2 )arctanx− 1
2 (x −arctan( x ))+C
¿ arctanx
2 (x2 +1)− x
2 +C
3.
4.

5. 1
6. ∫
0
π /2
sin x dx=¿−¿ ¿
7. .

= [ ln ( √ e2 x+ 4+ ex ) ]=ln ( √ e2 + 4+e ) −ln ( √ 5+1)

8. .
−1
2 ¿
9. ∫
x=−1
x=1
1−¿ x2 dx
2 =1
2 ∫ x− x3
3 dx= 1
2 [ 1− 1
3 +1−1
3 ] =2
3 ¿
10. −π
2 ∫ rdr=−π
2 [ r2
2 ]=−π
2 [ ∞ ]=∞
Ans 6. 1. In mathematics, the law of a stochastic process is the measure that the process induces on the
collection of functions from the index set into the state space.
2.
3. probability = 2/5 for x=15.