Integration by parts

In this question you must show detailed reasoning.

Use integration by parts to find the exact value of ln(3 - 2x)dx.

The diagram shows the curve with equation y = (x³ - 2x²)ln x. The curve has a point of inflection at the point M.

Show that the x-coordinate of M satisfies the equation

Use an iterative formula, based on the equation in part (a)(i), to determine the x-coordinate of M correct to 2 decimal places. Use an initial value of 1.1 and show the

result of each step of the iterative process.

Determine the exact area of the shaded region, giving your answer in the form p ln q - r,

where p and r are positive rational numbers and q is a positive integer.

Find the equation of this curve, giving your answer in the form e^3y = f(x).

Show that, when x = e², the y-coordinate of this curve can be written as

y = a + ln(be³ + c), where a, b and c are constants to be determined.

The diagram shows the curve M with equation y = xe^-2x.

Show that M has a point of inflection at the point P where x = 1.

The line L passes through the origin O and the point P. The shaded region R is enclosed by the curve M and the line L.

Show that the area of R is given by

where a and b are integers to be determined.

The diagram shows the curve C with parametric equations

The maximum point on C is denoted by P.

Determine the exact coordinates of P.

The shaded region R is enclosed by the curve, the x-axis and the lines x = 1 and x = 6.

Show that the area of R is given by

where a and b are constants to be determined.

Hence determine the exact area of R.

The diagram shows part of the curve y = ln (x - 4).

State the equation of the vertical asymptote to the curve y = ln (x - 4).

Find the total area enclosed by the curve y = ln (x - 4), the x-axis and the lines x = 4.5 x = 7. Give your answer in the form a ln 3 + b ln 2 + c where a, b and c are constants to be

found.