MAT1581 Assignment 4 (Accurate Solutions) Due 08 September 2025

MAT1581
Assignment 4
Due 08 September 2025

, MAT1581 Assignment 4 — Due date: 08 September 2025


MAT1581 Assignment 4
Due date: 08 September 2025


Solutions
Question 1: 34 Marks

10
R
(1.1) Evaluate 25−x2
dx.

Step 1: Use partial fractions.

10 1 1
= + .
25 − x2 5−x 5+x

Step 2: Integrate term by term.
Z Z
1 1
dx = − ln |5 − x|, dx = ln |5 + x|.
5−x 5+x

Final Answer: Z
10 5+x
2
dx = ln + C.
25 − x 5−x



R 2 −5
(1.2) Evaluate ln(e2x ) dx.

Z
ln e2x−5 dx




Step 1: Simplify the logarithm.
We use the identity ln(eu ) = u, so:

ln(e2x−5 ) = 2x − 5.


Step 2: Integrate.
Z Z Z
(2x − 5) dx = 2x dx − 5 dx.


Z Z
2
2x dx = x , −5 dx = −5x.

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