Exam (elaborations) maths

Integration
Satyabrat Sahu
February 2024


Integration Problem
Q1 Prove that:
5 Z π
X 4 cos x − sin x π3
logk 4 + 5π 2


I= dx =
−π cos x + sin x 32
k=1 4




√
n √
sin x − n cos x
Z
Q2 Ω(n) = √
n √ √ √ dx(n ∈ N, n ≥ 2)
( sin x + n
cos x)( 2n sin x + n cos x)

Q3 Find:
∞ ∞
log3 (x) log2 (x)
Z Z
3
Ω(a, b) = − log(ab) , a, b > 0
0 (x + a)(x + b) 2 0 (x + a)(x + b)


Q4 Prove the integral relation:
Z πs √ p√ !
csc y2 + π4


3 ( 4 2 + 1)( 2 + 2 − 1)
y

√ dy = 2 4 log √ p√
0 2 cos 2 + 2 ( 4 2 − 1)( 2 + 2 + 1)


Q5 Find:
"n+1   k1 # " n   1 #!−1
X 1 X 1 k+1
Ω = lim 1− · 1+ , [∗] − GIF
n→∞ k k
k=2 k=1



Q6 Prove that:
1
√ !
1 + 1 − x2
Z  
1+x 1
16G − π 2


Ω= log log √ dx =
0 1−x 1− 1−x 2 2


1