Integration by substitution

Use the substitution u = cos x to show that

Fig. 1 shows part of the curve y = The shaded region is bounded by the curve, the x-axis, and the line x = a.

You are given that the area of the shaded region is a square units.

Show that the value of a satisfies the equation a - cos^-1(1 - 2e^-a) = 0.

Show by calculation that the value of a lies between 1.1 and 1.2.

Use the iterative formula

a_n+1 = cos^-1(1 - 2e^-a_n),

with starting value a₁ = 1.2, to find the value of a correct to 2 decimal places. Show the

result of each step of the iterative process.

Fig. 2 shows the curve y = cos^-1(1 - 2e^-x).

Explain why the gradient of the curve y = cos^-1(1 - 2e^-x) at the point where x = a, where a

is the value found in part (d), lies in the interval (- 1, 0).

Use the substitution u = e^x - 2 to show that

Amaya and Ben integrated (1 + x)², with respect to x, using different methods, as follows.

Charlie said that, because these answers are different, at least one of them must be wrong.

Explain whether you agree with Charlie’s statement.

You are given that a is a constant greater than 1.

In this question you must show detailed reasoning.

In this question you must show detailed reasoning.

Determine the area of the shaded region, giving your answer in the form p + q ln 2 where p and q

are constants 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.

Use the trapezium rule, with two strips of equal width, to show that

Use the substitution x = u² to find the exact value of

Using your answers to parts (i) and (ii), show that

where k is a rational number to be determined.