MA1M01 Calculus Assignment 5

MA1M01 Calculus Assignment 5
Michælmas term week 8
www.maths.tcd.ie/pub/MA1M01/Calculus/
1. [30 points] Integrate each of the following by using a substitution:
(a)
Z
x4 (1 + x5 )3 dx
Z
ecos(x) sin(x)dx
Z 5 √
(e)
x3 x4 + 1dx
(c)
0
x3
dx
x4 + 12
Z0
2
(d) −2xe−x dx
Z
cos(x)
(f)
dx
(5 + sin(x))2
(b)
Z 1
√
2. [30 points] Integrate each of the following using integration by parts:
(a)
(c)
(e)
Z
Z
Z
x sin(x)dx
xe−x dx
2x2 ex dx
(b)
(d)
(f)
Z
x cos(10x)dx
Z
xe2x dx
Z
x2 ln(x)dx
3. [10 points] Without using a calculator solve for x:
(a) ln(2x3 ) − ln(x) = ln(16) + ln(x)
(b) ln(2x2 ) + ln(x) = ln(32) − ln(x)
4. [10 points] Without using a calculator simplify:
(a)
(b)
ln(32) + ln(8) − ln(4)
ln(256)
1
2
ln(16) − 13 ln(8)
ln(4)
Homework is due one week from when it is given in the tutorial you are assigned to.
This set should be handed up in week 9.
Marks may be lost if workings are not clearly shown.

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