Integration of x^n·ln(x)

4 Show that \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 3 } { 2 } } \ln x \mathrm {~d} x = 2 - \ln 4\).

2 Find the exact value of \(\int _ { 1 } ^ { 2 } x \ln x \mathrm {~d} x\).

3 Find the exact value of \(\int _ { 1 } ^ { 4 } \frac { \ln x } { \sqrt { } x } \mathrm {~d} x\).

3

1. Find \(\int \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x\).
2. Hence show that \(\int _ { 1 } ^ { 2 } \frac { \ln x } { x ^ { 3 } } \mathrm {~d} x = \frac { 1 } { 16 } ( 3 - \ln 4 )\).

3 Find the exact value of $$\int _ { 1 } ^ { 4 } x ^ { \frac { 3 } { 2 } } \ln x \mathrm {~d} x$$

2 Find the exact value of \(\int _ { 1 } ^ { 3 } x ^ { 2 } \ln 3 x \mathrm {~d} x\). Give your answer in the form \(a \ln b + c\), where \(a\) and \(c\) are rational and \(b\) is an integer. \includegraphics[max width=\textwidth, alt={}, center]{656df2a8-fc4d-49f3-a649-746103b4576e-04_2720_38_105_2010}

1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\)

Write your answer in the form \(a + \frac { b } { \mathrm { e } }\), where \(a\) and \(b\) are integers.

2. (a) Use integration to find $$\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$ (b) Hence calculate $$\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$

2 Evaluate \(\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x\), giving your answer in an exact form.

4

1. Differentiate \(\frac { \ln x } { x ^ { 2 } }\), simplifying your answer.
2. Using integration by parts, show that \(\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = - \frac { 1 } { x } ( 1 + \ln x ) + c\).

1 Find the exact value of \(\int ^ { 2 } x ^ { 3 } \ln x \mathrm {~d} x\).

3

1. Differentiate \(\frac { \ln x } { x ^ { 2 } }\), simplifying your answer.
2. Using integration by parts, show that \(\int \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x = - \frac { 1 } { x } ( 1 + \ln x ) + c\).

1 Evaluate \(\int _ { 1 } ^ { 2 } x ^ { 2 } \ln x \mathrm {~d} x\), giving your answer in an exact form.

2 Find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } x ^ { 4 } \ln x \mathrm {~d} x\).

3. Find the exact value of $$\int _ { 1 } ^ { 3 } x ^ { 2 } \ln x d x$$

3 Find the exact value of \(\int _ { 1 } ^ { 2 } x ^ { 3 } \ln x \mathrm {~d} x\).

3 Find \(\int _ { 1 } ^ { 4 } x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\), giving your answer in an exact form.

2 Find \(\int x ^ { 8 } \ln ( 3 x ) \mathrm { d } x\).

4 Find the exact value of \(\int _ { 1 } ^ { 8 } \frac { 1 } { \sqrt [ 3 ] { x } } \ln x \mathrm {~d} x\), giving your answer in the form \(A \ln 2 + B\), where \(A\) and \(B\) are constants to be found.

6 Find \(\int \frac { 32 } { x ^ { 5 } } \ln x \mathrm {~d} x\). Answer all the questions
Section B (78 marks)