MAT1581
Assignment 4
Due 08 September 2025
, Student Name: MAT1581 Assignment 4
MAT1581 Assignment 4
Due date: 08 September 2025
Question 1 (34 Marks)
Evaluate the following integrals. Each part below contains a short step-by-step
procedure.
(1.1)
Z
10
dx
25 − x2
Step 1: Recognise the standard integral
Z
1 1 a+x
dx = ln + C.
a2 − x 2 2a a−x
Step 2: Identify a2 = 25 ⇒ a = 5 and account for the factor 10:
Z
10 1 5+x
2
dx = 10 · ln + C.
25 − x 2·5 5−x
Final Answer:
5+x
ln +C
5−x
(1.2)
Z
ln e2x−5 dx
Step 1: Simplify the logarithm.
We use the identity ln(eu ) = u, so:
ln(e2x−5 ) = 2x − 5.
Page 1
Assignment 4
Due 08 September 2025
, Student Name: MAT1581 Assignment 4
MAT1581 Assignment 4
Due date: 08 September 2025
Question 1 (34 Marks)
Evaluate the following integrals. Each part below contains a short step-by-step
procedure.
(1.1)
Z
10
dx
25 − x2
Step 1: Recognise the standard integral
Z
1 1 a+x
dx = ln + C.
a2 − x 2 2a a−x
Step 2: Identify a2 = 25 ⇒ a = 5 and account for the factor 10:
Z
10 1 5+x
2
dx = 10 · ln + C.
25 − x 2·5 5−x
Final Answer:
5+x
ln +C
5−x
(1.2)
Z
ln e2x−5 dx
Step 1: Simplify the logarithm.
We use the identity ln(eu ) = u, so:
ln(e2x−5 ) = 2x − 5.
Page 1
